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DEMOGRAPHIC RESEARCH
A peer-reviewed, open-access journal of population sciences
DEMOGRAPHIC RESEARCH
VOLUME 48, ARTICLE 11, PAGES 321–338
PUBLISHED 28 FEBRUARY 2023
http://www.demographic-research.org/Volumes/Vol48/11/
DOI: 10.4054/DemRes.2023.48.11
Replication
The question of the human mortality plateau:
Contrasting insights by longevity pioneers
Linh Hoang Khanh Dang Carlo Giovanni Camarda
Nadine Ouellette France Mesl´
e
Jean-Marie Robine Jacques Vallin
©2023 Dang, Camarda, Ouellette, Mesl´
e, Robine & Vallin.
This open-access work is published under the terms of the Creative
Commons Attribution 3.0 Germany (CC BY 3.0 DE), which permits use,
reproduction, and distribution in any medium, provided the original
author(s) and source are given credit.
See https://creativecommons.org/licenses/by/3.0/de/legalcode
Demographic Research: Volume 48, Article 11
Replication
The question of the human mortality plateau:
Contrasting insights by longevity pioneers
Linh Hoang Khanh Dang1
Carlo Giovanni Camarda2
Nadine Ouellette3
France Mesl´
e4
Jean-Marie Robine5
Jacques Vallin6
Abstract
BACKGROUND
The debate about limits to the human life span is often based on outcomes from mor-
tality at the oldest ages among longevity pioneers. To this day, scholars disagree on
the existence of a late-life plateau in human mortality. Amid various statistical analy-
sis frameworks, the parametric proportional hazards model is a simple and valuable ap-
proach to test the presence of a plateau by assuming different baseline hazard functions
on individual-level data.
OBJECTIVE
We replicate and propose some improvements to the methods of Barbi et al. (2018) to
explore whether death rates reach a plateau at later ages in the French population as it
does for Italians in the original study.
METHODS
We use a large set of exceptionally reliable data covering the most recently extinct birth
cohorts, 1883–1901, where all 3,789 members who were born and died in France, were
followed from age 105 onward. Individual life trajectories are modeled by a proportional
1Institut National d’Etudes D´
emographiques, France. Email: danghoangkhanhlinh@gmail.com.
2Institut National d’Etudes D´
emographiques, France.
3Department of Demography, Universit´
e de Montr´
eal, Canada.
4Institut National d’Etudes D´
emographiques, France.
5Institut National de la Sant´
e et de la Recherche M´
edicale, and ´
Ecole Pratique des Hautes ´
Etudes, France.
6Institut National d’Etudes D´
emographiques, France.
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Dang et al.: The question of the human mortality plateau: Contrasting insights by longevity pioneers
hazards model with fixed covariates (gender, birth cohort) and a Gompertz baseline haz-
ard function.
RESULTS
In contrast with Barbi et al. (2018)’s results, our Gompertz slope parameter estimate is
statistically different from zero across all model specifications, suggesting death rates
continue to increase beyond 105 years old in the French population. In addition, we find
no significant birth cohort effect but a significant male disadvantage in mortality after age
105.
CONCLUSIONS
Using the best data currently available, we did not find any evidence of a mortality plateau
in French individuals aged 105 and older.
CONTRIBUTION
The evidence for the existence of an extreme-age mortality plateau in recent Italian co-
horts does not extend to recent French cohorts. Caution in generalizations is advised, and
we encourage further studies on long-lived populations with high-quality data.
1. Introduction
The shape of the mortality curve at very old ages has long been a subject of heated de-
bate, with opinions divided mainly on whether death rates at the most advanced ages
continue to increase exponentially or at a slower pace (i.e., decelerate). The latter, if true,
leads to another possible scenario, where death rates not only decelerate but eventually
become constant after a certain age, reaching the so-called ‘plateau of human mortal-
ity’. Acknowledging the mortality plateau means validating the existence of mortality
deceleration in humans, but the opposite is not necessarily true.
Our ability to study mortality deceleration and the existence of a mortality plateau
depends chiefly on having high-quality data on death counts and population size at very
high ages, where we can expect to observe these two phenomena. The methods used
to analyze the trustworthy data also matter and should be adapted to the type of data at
hand (censored, truncated, period- or cohort-based, individual- or aggregated-level, etc.).
Different combinations of data and methods continue to fuel the long-standing debate,
provoke controversies, and make it difficult to reach a single conclusion. For instance,
several works published during the late 1990s based on aggregated period mortality data
from the 1950s onward documented the deceleration in the age pattern of mortality at
advanced ages in a large set of low-mortality countries (Horiuchi and Wilmoth 1998;
Thatcher, Kannisto, and Vaupel 1998; Thatcher 1999). More recently, mortality decel-
eration was also observed in a thoroughly validated set of data on French-Canadian cen-
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Demographic Research: Volume 48, Article 11
tenarians born to (extinct) cohorts 1870–1896 (Ouellette 2016), as well as thanks to an
extensive testing of mortality models after age 80 on 360 high-quality cohort data sets
(Feehan 2018). Evidence of such a late-life mortality deceleration seems consistent with
various theoretical explanations (Beard 1959; Vaupel et al. 1979; Wachter and Finch
1997; Wilmoth and Horiuchi 1999; Steinsaltz and Wachter 2006; Mueller at al. 2011).
Still, no consensus has yet been reached because other scholars have found that in some
period and cohort settings, the death rate increases exponentially up to at least ages 105 to
106 (Gavrilov and Gavrilova 2011; Gavrilova and Gavrilov 2015; Gavrilov and Gavrilova
2019b; Gavrilova, Gavrilov, and Krut’ko 2017).
In this article, we focus on a possible mortality plateau in humans. The last 15 years
have yielded more and more reliable individual-level data on deaths at the latest ages,
thereby creating new research opportunities. Gampe (2010) for instance implements a
nonparametric approach and estimated a mortality plateau at a (constant) hazard level of
about 0.7 for females after age 110, using individual-level data from the 2008 edition
of the International Database of Longevity (IDL) on 11 countries. A decade later, the
updated version of the IDL, which included roughly twice as many individuals as previ-
ously, permits a re-analysis of the late-age trajectory of human mortality and confirms the
constant hazards of death above age 110 (Gampe 2021). The study also renews warning
against drawing any conclusion beyond the age of 114 due to data scarcity. Meanwhile,
in 2018, the journal Science had released a paper by Barbi and colleagues in which the
authors use newly available Italian individual-level data on survivors and deaths above
age 105 to claim evidence for the “existence of extreme-age mortality plateaus in hu-
mans” (p. 1459). The latter study generates a great debate, with opinions ranging from
agreement (Mueller and Rose 2018), to reservations (Camarda et al. 2018), to opposi-
tion (Beltr´
an-S´
anchez, Austad, and Finch 2018; Olshansky and Carnes 2018; Newman
2018; Gavrilov and Gavrilova 2019a), fueling long-standing discussions on the possible
limits to human life span, where various means of statistical analysis have been applied
to different sources of data (Aarssen and de Haan 1994; Barbi, Caselli, and Vallin 2003;
Dong, Milholland, and Vijg 2016; Gbari et al. 2017; Vijg and Le Bourg 2017; Rootz´
en
and Zholud 2017; Einmahl, Einmahl, and de Haan 2019). We find Barbi and colleague’s
(2018) approach interesting and worthy of replication on our large set of exceptionally
high-quality data, consisting of 3,789 French longevity pioneers born in France between
1883 and 1901, each followed from age 105 until death.
2. Data
Our data come from the R´
epertoire national d’identification des personnes physiques
(RNIPP), a database maintained by the French National Institute for Statistics and Eco-
nomic Studies (Institut national de la statistique et des ´
etudes ´
economiques, or INSEE).
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Dang et al.: The question of the human mortality plateau: Contrasting insights by longevity pioneers
RNIPP data include a list of individual records linking each person to an identification
number, thus identifying specific individuals without error. In 2014, INSEE signed an
agreement with the French Institute for Demographic Studies (Institut national d’´
etudes
d´
emographiques, or INED), whereby INSEE provides INED with an RNIPP extract of all
records of individuals born in France whose differences between alleged years of death
and birth are at least 105. INSEE updates this extract for INED on a yearly basis, which
allows for continuous empirical studies on mortality at extreme ages in France.
Table 1: Number of validated deaths by birth cohort and sex, French
population, ages 105 and above
Birth cohort Number of deaths
Females Males
1883 71 10
1884 73 5
1885 79 11
1886 95 10
1887 124 8
1888 123 11
1889 148 13
1890 142 15
1891 164 13
1892 150 14
1893 204 22
1894 207 21
1895 223 12
1896 240 21
1897 264 19
1898 279 20
1899 236 20
1900 308 27
1901 355 30
Total 3,485 304
Source: R ´
epertoire national d’identification des personnes physiques (RNIPP).
Due to its long history of well-developed civil registrations, France is one of the few
countries that have gathered comparably vast amounts of data on deaths at ages 105 and
above. For the sake of population homogeneity and accuracy, we included in our dataset
only individuals who were born and died in metropolitan France, excluding all cases of
deaths that occurred outside the country, as well as births and/or deaths pertaining to
French overseas departments and territories (so-called DOM–TOM). Moreover, we kept
data on extinct birth cohorts exclusively to follow all members of the various cohorts
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Demographic Research: Volume 48, Article 11
from age 105 until extinction (i.e., up to age 115 in the present study). Therefore, we
have neither left-truncated nor right-censored data in our analyses, but right truncation
exists, and our models will take it into account. These data are available in the IDL
(2022), where data are provided free of charge after registration.
The distribution of deaths by birth cohort and sex is presented in Table 1. Altogether,
there are 19 extinct cohorts born between 1883 and 1901, covering 3,789 deaths above age
105 that occurred during the 1988–2016 period. Females outnumber males considerably
at the oldest ages (3,485 vs. 304), and the number of female survivors also increases
steadily across birth cohorts.
Since data quality at the oldest ages depends heavily on the accuracy of reported
ages at death, the data we use here were validated following a strict protocol for thor-
oughly verifying the coherence between the information recorded on the person’s official
birth and death certificates. The details and results of the age-validation procedure on
the French dataset can be found in Ouellette et al. (2021). In summary, for deaths that
occurred at ages 110 onward (so-called supercentenarians), exhaustive validation was
performed. All birth and death certificates were recovered and confirmed the accuracy
of age at death (to the nearest day) for the vast majority of cases: 92% of the 213 cases
were correct (i.e., 18 erroneous cases were found). For deaths registered at ages 105 to
109 (so-called semi-supercentenarians), which are quite numerous, the validation process
was applied to a set of extracted RNIPP data that covers 1,050 deaths from years 1988 to
2002. This set was checked exhaustively for the oldest individuals (alleged ages of 107,
108, and 109). For the remainder of the set, a random sample made of half the cases at
alleged age 106 and one third of those at alleged age 105 was drawn and checked. All
birth and death certificates for these French alleged semi-supercentenarians were recov-
ered (except for 4 individuals), and the registered ages had a very high degree of accuracy:
99.7% of the 2,031 cases were correct (i.e., only 3 erroneous cases). We therefore con-
cluded that these most recent data from the RNIPP extracts require no further validation
for ages 105 to 109. Globally, according to the IDL – which collates data on deaths at
ages 105 and above from countries with reliable civil registration (or equivalent) systems
– the French data validation procedure meets the highest criteria.
3. Methods
Individual life trajectories leading to death provide continuous information over time
without requiring data to be aggregated by age and birth cohort. We model these tra-
jectories from age 105 until death using a parametric proportional hazards model with
fixed covariates (gender and birth cohort), in which the baseline hazard function follows
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Dang et al.: The question of the human mortality plateau: Contrasting insights by longevity pioneers
the Gompertz model (Gompertz 1825). The formula for the individual hazard is given by
h(ti) = h0(ti) exp(β1Ci+β2Mi) = aexp(b ti) exp(β1Ci+β2Mi) ,
where Ciis the individual’s birth year minus 1891, Mi= 1 for males and 0 for females,
and tiis the survival duration (in years) of each individual after age 105.
Consequently, β1and β2capture the cohort and gender effects, respectively; ais the
initial hazard at starting age 105; and bis the Gompertz slope. The baseline hazard func-
tion h0(ti)can thus be interpreted as the hazard for female subjects born in 1891. With
this type of model, a straightforward test can establish the significance of each parameter
and evaluate the effect of explanatory variables. For instance, β2measures the difference
in hazards between males and females, given equal birth cohorts and controlling for age.
The vector of parameters θ= (a,b,β1,β2)′is estimated using the method of maxi-
mum likelihood. For individuals who have survived to the age of 115 or beyond our last
date of observation, the right-truncated duration, R, is the difference between that last
date of observation and the date on which these individuals attained age 105 (0< t < R).
Given θ, the contribution to the likelihood function is
L(t,R;θ) = f(t;θ)
F(R;θ),
where f(·)and F(·)are the density and distribution functions, respectively.
Taking logarithms on both sides of the previous equation and applying survival ana-
lysis relationships, we obtain the contribution to the log-likelihood function
ln L(t,R;θ) = ln(h(t;θ)–H(t;θ)–ln(1 –S(R;θ)),
where h(·)and H(·)are the hazard and cumulative hazard functions evaluated at the
duration lived, respectively. The last term, S(R;θ), is the survival function at the time of
right truncation. For further details on these survival analysis derivations, see for example
Hosmer and Lemeshow (1998) and Klein and Moeschberger (2003).
In this article, maximum likelihood estimation is performed using the Rpackage
flexsurv (Jackson 2016: v2.0), which has an option to take into account the right
truncation present in our data scheme. Reproducible Rcode is provided with this paper,
and readers are invited to refer to these materials for more details.
A major objective in our study is to test the hypothesis of constant mortality above
age 105 against the hypothesis that mortality continues to increase after this age. In
order to reproduce the analysis presented in Barbi et al. (2018), we first focused on the
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Demographic Research: Volume 48, Article 11
hypothesis testing performed in the original article where
H0:b= 0 vs.H1:b= 0 .
This type of test is, however, unsuitable when dealing with survival data because H1:
b= 0 allows for eventual negative values of b, which in the Gompertz model leads to a
defective distribution where a fraction of the individuals could live forever.
Consequently, we decided to go beyond the mere purpose of replicating the Barbi
et al. (2018) analysis on our high-quality data of the French population aged 105 and
above. Hence, we also perform a more suitable hypothesis testing where b > 0is opposed
to the null hypothesis of a mortality plateau, which is
H0:b= 0 vs.H1:b > 0 .
In this case, parameter bis on the boundary of the parameter space, which makes the stan-
dard assumptions on asymptotic properties of likelihood-based inference inappropriate.
Therefore, the likelihood ratio test should be done using another asymptotic distribution
of the likelihood ratio test statistic. In a demographic setting, Bohnstedt and Gampe
(2019: p. 71) show that we should use a mixture of a chi-squared distribution with one
degree of freedom and a point mass at 0, given by 1
2χ2
1+1
2χ2
0.
We ran statistical hypothesis tests on all model specifications to assess the relevance
of gender and cohort effects. Associated coefficients are tested by opposing β= 0 to
β= 0 (i.e., by performing a conventional likelihood ratio test as models are nested).
The performance of all possible models can be compared using the Akaike infor-
mation criteria (AIC) (Akaike 1973). This criterion, which balances a model’s fidelity to
data and complexity, is computed as
AIC = –2L+ 2k,
where Lis the maximized likelihood value and kis the number of parameters estimated
for each model. The absolute values of AIC are not interpretable per se, but the difference
between AIC scores allows us to assess the associated models’ fit while accounting for
the number of estimated parameters. The smaller the AIC score, the better the model
performs.
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Dang et al.: The question of the human mortality plateau: Contrasting insights by longevity pioneers
4. Results
4.1 Constant vs. Gompertz mortality after age 105
To assess the significance of the additional Gompertz slope, b, with respect to a constant
mortality scenario, we performed two statistical hypothesis tests in which parameter b=
0is opposed to either b= 0 or to a strictly positive b. The former approach allows us
to replicate the test conducted by Barbi et al. (2018) using our French dataset, while the
latter is more appropriate for our setting by avoiding immortal individuals (see Section 3
for details).
Table 2 presents the outcomes of the two hypothesis tests on all model specifications.
Interaction effects are not considered, as in the original study by Barbi et al. (2018). In
any case, these interactions were not statistically significant in our data. For every model
specification, a likelihood ratio test rejects the null hypothesis of a constant hazard after
age 105. This outcome holds regardless of the hypothesis test performed. In fact, by
imposing a nonnegative constraint on parameter b, the test becomes one- rather than two-
tailed and p-values are consequently halved, giving an even higher level of significance
for the Gompertz slope parameter in all our model specifications.
Table 2: p-values for two hypothesis tests on Gompertz slope parameter, b,
across all model specifications, French birth cohorts 1883–1901,
ages 105 and above
Models p-values: H0:b= 0 v s .
H1:b>0H1:b= 0
No covariate 7.011e-08 1.402e-07
With gender effect 4.389e-08 8.778e-08
With cohort effect 7.063e-08 1.413e-07
With gender and cohort effects 4.380e-08 8.761e-08
Source: Authors’ calculations based on data from the R ´
epertoire national d’identification des personnes physiques
(RNIPP).
A comparison based on AIC is presented in Table 3, and it also shows that the model
with the Gompertz slope, b, describes the data better (e.g., 10,878.00 vs. 10,904.63 for the
model with gender effect only). In sum, based on the reported parameter estimates, on the
hypothesis tests (one- and two-sided) and on the AIC, the Gompertz slope, b, is highly
significant in the French data. To put it another way, the observed data is sufficiently
inconsistent with the null hypothesis of a constant mortality pattern.
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Demographic Research: Volume 48, Article 11
Table 3: Parameter estimates in model specifications with different
combinations of variables, French birth cohorts 1883–1901,
ages 105 and above
Baseline
Covariates Parameter
estimates 95% CI Log-
likelihood AIC Rank
No covariates a0.645 [ 0.625 , 0.666 ] -5,452.094 10,906.19 7
a0.645 [ 0.621 , 0.670 ]
Cohort β1-0.00035 [-0.0065, 0.0058] -5,452.087 10,908.17 8
a0.638 [ 0.617 , 0.659 ]
Sex β20.144 [ 0.027 , 0.261 ] -5,449.319 10,902.64 5
a0.645 [ 0.614 , 0.664 ]
β10.000 [-0.006 , 0.006 ]
Constant hazard, a
Cohort+
Sex β20.144 [ 0.026 , 0.261 ]
-5,449.317 10,904.63 6
a0.589 [ 0.562 , 0.617 ]
No covariates b0.061 [ 0.039 , 0.083 ] -5,438.235 10,880.47 3
a0.588 [ 0.559 , 0.619 ]
b0.061 [ 0.039 , 0.084 ]
Cohort
β10.000 [-0.006 , 0.006 ]
-5,438.234 10,882.47 4
a0.581 [ 0.554 , 0.609 ]
b0.062 [ 0.040 , 0.084 ]
Sex
β20.155 [ 0.038 , 0.273 ]
-5,435.006 10,876.01 1
a0.580 [ 0.550 , 0.612 ]
b0.062 [ 0.040 , 0.084 ]
β10.00028 [-0.006 , 0.007 ]
Gompertz (a,b)
Cohort+
Sex
β20.156 [ 0.038 , 0.273 ]
-5,435.002 10,878.00 2
Notes: CI is for confidence interval, AIC is for Akaike information criterion. Letters aand brefer to Gompertz
parameters from the model h(t) = aexp(b t);β1and β2capture cohort and sex effect, respectively.
Source: Authors’ calculations based on data from the R ´
epertoire national d’identification des personnes physiques
(RNIPP).
To assess departure from the data due to the parametric assumption of the baseline,
we also computed the cumulative hazard using both nonparametric and parametric ap-
proaches, following the idea of Barbi and colleagues (2018). For the former, since right
truncation is present in our study framework, instead of using the conventional Kaplan-
Meier estimator, we relied on the estimator proposed by Shen (2010) and implemented
in the function cdfDT provided by the Rpackage SurvTrunc (Rennert 2018). A small
modification was made to the code of the cdfDT function to extract the cumulative haz-
ard estimates needed for this visual inspection. As done for the parametric approach, we
computed the cumulative hazard from both the constant and Gompertz hazards accord-
ing to the relationship H(t) = –ln[S(t)] and using the parameter estimates in Table 3
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Dang et al.: The question of the human mortality plateau: Contrasting insights by longevity pioneers
for models including a statistically significant gender effect. Figure 1 presents estimates
for the French female cohort, and it shows that the Gompertz hazard model (blue line)
is closer to the nonparametric estimates (in red) than the fitted constant hazard (green
line). The gap between the two lines (blue and green) becomes more noticeable as age
increases, and at the highest ages, the cumulative hazard curve under constant hazard
(green line) is often found to lie below the lower bound of the nonparametric confidence
interval estimates. These observations strengthen our finding that the Gompertz baseline
performs best in modeling hazard rates above age 105.
Figure 1: Estimated cumulative hazard using nonparametric and
parametric approaches, French females born 1883–1901,
ages 105 and above
Age (years)
Cumulative hazard
105 106 107 108 109 110 111 112 113 114
0
1
2
3
4
5
6
7
8
NPMLE estimates & 95% CIs
Under Gompertz hazard
Under constant hazard
Notes: NPMLE is for nonparametric maximum likelihood estimate. This figure is equivalent to Figure 2 in Barbi
et al. (2018: p. 1461).
Source: Authors’ calculations based on data from the R ´
epertoire national d’identification des personnes physiques
(RNIPP).
Going one step further, we consider that a useful model should not only fit the data
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Demographic Research: Volume 48, Article 11
well over a specific age range (i.e., from age 105 onward here), but it should also be able
to summarize the transition from younger old ages as smoothly as possible. As natura
non facit saltus, the force of nature upon human mortality shall change gradually in ab-
sence of specific reasons for irregularities. To check for conformity, we plotted in Figure
2 observed death rates at ages 90 to 104 taken from the Human Mortality Database (2020)
for the cohort of French females and males born in 1891. We then superposed the asso-
ciated sex-specific fitted curves from age 105 given by our models with a Gompertz and
a constant hazard baseline. Despite the relatively large confidence intervals (shaded pur-
ple area) due to small population sizes, especially beyond age 110, a Gompertz baseline
aligns better with the mortality trend at younger old ages. This confirms once more that a
Gompertz baseline function is more reasonable for our data when modeling hazard rates
above age 105.
Figure 2: Observed and fitted hazards, French females (left) and males
(right), born in 1891
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
90 95 100 105 110 115
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Age (years)
Hazard (log scale)
●
●
●
●
●●
●
●
●
●
●
HMD data Validated
individual data
Gompertz PH
Constant PH
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
90 95 100 105 110 115
Age (years)
●
●
●
●
●
●
●
HMD data Validated
individual data
Gompertz PH
Constant PH
Notes: PH is for proportional hazards. Associated confidence intervals do not account for right truncation scheme
and are included for illustrative purposes only. This figure is equivalent to Figure 1 in Barbi et al. (2018: p. 1461).
Source: Authors’ calculations based on data from the R ´
epertoire national d’identification des personnes physiques
(RNIPP) and Human Mortality Database (HMD).
4.2 Cohort and gender effects on mortality after age 105
We investigated other model specifications, first to test the sensitivity of our parameter
estimates and second to choose the best-performing model for reliable conclusions on
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Dang et al.: The question of the human mortality plateau: Contrasting insights by longevity pioneers
birth cohort and gender effects. The results across all model specifications are reported in
Table 3. Precisely, we performed likelihood ratio tests between models having different
baseline functions without cohort or gender effects (p= 1.402e–07, see Table 2), be-
tween models with gender effect only (p= 8.778e–08), and between models with cohort
effect only (p= 1.413e–07). We find that the Gompertz slope parameter, b, is posi-
tive and statistically different from zero regardless of the model specification (Table 3).
According to the AIC, models including the Gompertz slope consistently perform better
than those assuming constant hazard. Among them, the model with a Gompertz baseline
hazard and a gender covariate has the lowest AIC, thereby showing that gender still has a
significant effect on mortality after age 105. Assuming all else being equal, male subjects
have a hazard rate that is 1.168 times higher than their female counterparts (hazard ratio
of exp(0.155) = 1.168). Hence, according to our data for France, a male disadvantage
persists even at the highest ages. No significant cohort effect is detected in any of the
model specifications.
5. Discussion
Our replication study is based on Barbi et al. (2018)’s paper, in which the authors rely
on an Italian dataset that appears to support the existence of a mortality plateau in hu-
mans beyond 105 years old. We investigated the generalizability of the original study by
using data for France and found no evidence of a plateau after the age of 105. Because
the methods in both studies are the same, explanations for the contradictory results likely
come from the data. The Italian and French datasets are similar in terms of sample size
(3,836 vs. 3,789) and hence of comparable statistical power. The latter set includes in-
dividuals who were born and died in metropolitan France only, while the former has a
small share of individuals born outside Italy (under 4% according to the authors). The
age-validation procedure for both sets of data consists in coupling birth and death certifi-
cates for all native alleged supercentenarians (110 and above). For younger old ages, the
French data go a little further as this coupling procedure was extended to a large number
of alleged semi-supercentenarians (105 to 109 years). With 99.7% of the 2,031 cases
checked having exact dates of birth and death, the very high quality of France’s RNIPP
data is confirmed.
Another key advantage of the French dataset is that all individuals belong to ex-
tinct birth cohorts. Unlike in the original study, these data are thus free from any left
truncation and right censoring, minimizing sources of uncertainty in mortality models’
parameter estimates. After careful examination, the right truncation, though theoretically
present in our data, does not have much of an effect on the parameter estimates in the
extinct-birth-cohorts data scheme. Our estimated value of the Gompertz slope parameter,
b, is much higher than that reported by Barbi et al. (2018) (0.062 vs. 0.013). With a rela-
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Demographic Research: Volume 48, Article 11
tively comparable sample size, a higher level of bgrants higher chances of detecting the
true trajectory instead of the simpler constant baseline hazard model that supports the ex-
istence of a mortality plateau. The low estimate of bin the study using Italian data might
be one of the reasons why constant baseline hazard model is selected even if the true
underlying mortality pattern could have been more likely Gompertzian (Camarda 2022).
We considered the same covariates as those of the original study, namely gender and
birth cohort. Despite the small number of male survivors at extreme ages (N= 304), we
found a significant gender effect on mortality after age 105 at the 5% level, indicating a
persistent male disadvantage even at the highest observed ages. Being male increases the
risk of death by a factor of 1.168 compared to females, all else being equal. Males having
a higher hazard than females of equal age agrees with the observation made by Barbi and
colleagues (2018). However, the increased risk of death for males, as indicated by the
gender parameter estimated from the Italian dataset, is of smaller magnitude compared
to ours (1.034 vs. 1.167) and did not lead to a significant gender effect at the 5% level
(p= 0.058) (Barbi et al. 2018: Table 2). Regarding the birth cohort covariate, it showed
no significant effect based on our French data, as opposed to what was found in the Italian
dataset. In addition to the fact that French birth cohorts are generally older than that of
Italy (1883–1901 vs. 1896–1910), the lack of a cohort effect in our analysis might also
come from the absence of clear mortality reductions after ages 90 to 92 in the French
population (Mesl´
e and Vallin 2020). It is therefore probably still too early to observe any
mortality improvement among semi- and supercentenarians in our birth cohorts, hence the
lack of a cohort effect. At this point, these are solely possible explanations, and since no
underlying mechanism has yet been confirmed, it is therefore worthy of further research.
The choice of the age range over which modeling is performed in our study (105
and above) was made to fully replicate the study by Barbi and colleagues (2018). It is,
however, an arbitrary choice, and it may have a direct effect on our modeling of age
patterns in mortality at the oldest ages. Although we found no sign of a mortality plateau
beyond age 105 in the French dataset, we cannot rule out the possibility of such a plateau
at later ages that our dataset does not allow us to study. Nor do we dismiss the possibility
of mortality deceleration above age 105; models are those of the original study and they
consider either constant or linear (Gompertz) mortality trends, not decelerating mortality
patterns.
6. Conclusion
Using a standard survival analysis procedure on French data that comprise all individual
deceased above age 105 in extinct birth cohorts ranging from 1883 to 1901, we found no
evidence of a plateau in human mortality after 105. Our estimate of the (baseline) Gom-
pertz slope parameter, b, is statistically different from zero across all model specifications
http://www.demographic-research.org 333
Dang et al.: The question of the human mortality plateau: Contrasting insights by longevity pioneers
studied. Such a finding suggests the Gompertz baseline is valid within a proportional
hazards framework and that death rates continue to increase beyond 105 years. This dis-
agrees with what Barbi and colleagues (2018) find in their study, which our paper aimed
to replicate by using French instead of Italian mortality data.
While many controversies have arisen as a result of different combinations of data
and methods, every new study should be seen as providing additional insight into the con-
tinuing debate about the shape of the mortality curve at very old ages, rather than granting
a final universal answer valid for all human populations. With each improvement in data
quality, increase in population size, and advance in statistical methodology, it is highly
probable that new outcomes will emerge. At present, however, it would be more prudent
to admit that no definitive conclusion about the existence of a plateau in human mortal-
ity at very high ages can be reached. Nevertheless, the advancing frontier of survival
will likely come to a point where the population surviving to current highest ages will
no longer be considered ‘pioneers’, and consequently there will be more solid knowledge
about mortality at these extreme ages. Until then, comprehensive and comparative studies
based on longevity data from different countries ought to be most welcome.
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Demographic Research: Volume 48, Article 11
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