A modal age at death approach to forecasting adult mortality

Abstract

Recent studies have shown that there are some advantages to forecasting mortality with indicators other than age-specific death rates. The mean, median, and modal ages at death can be directly estimated from the age-at-death distribution, as can information on lifespan variation. The modal age at death has been increasing linearly since the second half of the twentieth century, providing a strong basis from which to extrapolate past trends. The aim of this paper is to develop a forecasting model that is based on the regularity of the modal age at death and that can also account for changes in lifespan variation. We forecast mortality at ages 40 and above in 10 West European countries. The model we introduce increases forecast accuracy compared with other forecasting models and provides consistent trends in life expectancy and lifespan variation at age 40 over time.

Full text

A modal age at death approach to forecasting
adult mortality
Marie-Pier Bergeron-Boucher , Paola Vázquez-Castillo
and Trifon I. Missov
Interdisciplinary Centre on Population Dynamics, University of Southern Denmark
Recent studies have shown that there are some advantages to forecasting mortality with indicators other
than age-specific death rates. The mean, median, and modal ages at death can be directly estimated from
the age-at-death distribution, as can information on lifespan variation. The modal age at death has been
increasing linearly since the second half of the twentieth century, providing a strong basis from which to
extrapolate past trends. The aim of this paper is to develop a forecasting model that is based on the
regularity of the modal age at death and that can also account for changes in lifespan variation. We
forecast mortality at ages 40 and above in 10 West European countries. The model we introduce
increases forecast accuracy compared with other forecasting models and provides consistent trends in
life expectancy and lifespan variation at age 40 over time.
Supplementary material for this article is available at: https://doi.org/10.1080/00324728.2024.2310835
Keywords: mortality; forecast; modal age at death; lifespan variation; life expectancy
[Submitted November 2022; Final version accepted October 2023]
Introduction
The rise in life expectancy over the last two centuries
is one of the most remarkable achievements of
human populations. Life expectancy at birth was
around 40 years in the middle of the nineteenth
century and reached 87 years for females in Japan
in 2020 (Oeppen and Vaupel 2002; HMD 2022).
This constant increase has led to important demo-
graphic and societal changes, such as population
growth and ageing. Due to these continuing mor-
tality changes and related consequences, public and
private institutions rely on mortality forecasting to
anticipate healthcare needs and pension costs, for
example. The last few decades have witnessed an
important increase in the number of forecasting
models.
One model recurrently used for forecasting
mortality is the Lee–Carter (LC) model (Lee and
Carter 1992), which forecasts age-specific death
rates log-bilinearly. The advantages of the LC
model include its simplicity, the limited subjective
judgement required, and its direct forecasts of the
risk of death over time. However, the accuracy of
this model is often questioned. The LC model
assumes a constant rate of mortality improvement
in age-specific death rates over time, but evidence
shows that there has been accelerated mortality
decline at older ages in some populations (Rau
et al. 2008; Vaupel et al. 2021). Thus, the assumption
often leads to the method under-predicting life
expectancy (Lee and Miller 2001; Bergeron-
Boucher and Kjærgaard 2022). Variants of the LC
model have been suggested over the years, to
improve the accuracy of the original model (Booth
et al. 2002; Booth et al. 2006; Li and Lee 2005;
Hyndman and Ullah 2007; Currie 2013; Li et al.
2013; Camarda and Basellini 2021).
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Population Studies, 2025
Vol. 79, No. 1, 27–43, https://doi.org/10.1080/00324728.2024.2310835

Many forecasting models are based on the extrapo-
lation of age-specific death rates, as these rates
are indicative of the change in the risk of dying
over time and, in addition, serve as the point of
entry to the life table (Basellini and Camarda
2019). However, mortality forecasts can use a
number of other meaningful demographic indicators
as inputs (Bergeron-Boucher et al. 2019a). Some
authors have suggested forecasting age-specific
death probabilities (Cairns et al. 2006; King and
Soneji 2011). Using either age-specific death rates
or probabilities as inputs to the same model gener-
ally leads to similar forecasted trends (Bergeron-
Boucher et al. 2019a). Other input indicators, such
as life expectancy (Raftery et al. 2012; Torri and
Vaupel 2012; Pascariu et al. 2018) and the age-at-
death distribution (Oeppen 2008; Bergeron-
Boucher et al. 2017; Basellini and Camarda 2019),
have gained popularity, as they allow for changes
in age-specific rates of mortality improvement
(ASRMI).
Using life expectancy as the input has the
advantage of directly forecasting the average dur-
ation of life, and it is the most popular measure of
longevity. Life expectancy, as an aggregate indicator,
is less volatile than age-specific measures of mor-
tality. The associated forecasting models are not
only more robust but also more parsimonious
(Bergeron-Boucher et al. 2019a). However, life
expectancy does not provide any information about
age-specific mortality trends and levels, and
researchers must rely on an additional model to
derive age-specific mortality from the life expect-
ancy values (Ševc
íková et al. 2016; Pascariu et al.
2020; Nigri et al. 2022). As such, if information on
death rates or probability of survival to certain
ages (e.g. age at retirement) is needed, models
based on life expectancy lose their parsimony.
The age-at-death distribution readily provides
information on central longevity measures: the
mean, median, and mode (Canudas-Romo 2010).
While the mean (life expectancy) is the most used
measure, the mode is increasingly seen as an alterna-
tive measure of longevity (Canudas-Romo 2008;
Horiuchi et al. 2013). The modal age at death, M,
is the age at which the most adult deaths occur. In
low-mortality countries, M has been increasing
since the 1930s–40s for females and since the 1970s
for males (Canudas-Romo 2010; Bergeron-Boucher
et al. 2015; Diaconu et al. 2016). An increase in the
modal age at death indicates that the age-at-death
distribution is shifting towards older ages, a
dynamic interpreted as postponement of the mor-
tality schedule. The latter, also referred to as shifting
mortality, has been the dominant determinant of life
expectancy increase in low-mortality countries since
the middle of the twentieth century (Canudas-Romo
2008; Bongaarts 2009; Bergeron-Boucher et al.
2015). Increases in the modal age at death are
driven primarily by decreases in mortality at older
ages (Horiuchi et al. 2013; Diaconu et al. 2020),
especially at ages above the mode (Canudas-Romo
2010), making this indicator particularly relevant
for studying longevity extension.
Additionally, the age-at-death distribution pro-
vides information about the variability of lifetimes.
This measure, also called lifespan variation, captures
inequalities in lifespans from a population perspec-
tive. Lifespan variation can be directly calculated
from the age-at-death distribution and is increasingly
seen as a relevant complement to life expectancy and
the modal age at death (Tuljapurkar 2001; Vaupel
et al. 2011; van Raalte et al. 2018). Lifespan variation
has been decreasing in most low-mortality popu-
lations since the late nineteenth century (Edwards
and Tuljapurkar 2005), and this has led to a com-
pression of mortality around the mode. Unlike shift-
ing mortality, mortality compression (or reduction in
lifespan variation) is driven by a reduction in mor-
tality at younger ages (Aburto et al. 2020). There
is, generally, a negative correlation between life
expectancy and lifespan variation, as both measures
are sensitive to mortality changes at younger ages.
However, this is not a mechanical relationship, and
many exceptions where lifespan variation increases
with life expectancy have been documented (van
Raalte et al. 2011; Brønnum-Hansen 2017; Aburto
and van Raalte 2018; van Raalte et al. 2018).
The age-at-death distribution is increasingly used
as an input to mortality forecasting models
(Oeppen 2008; Bergeron-Boucher et al. 2017;
Basellini and Camarda 2019; Aliverti et al. 2022;
Shang et al. 2022). Basellini and Camarda (2019)
introduced an innovative model to forecast the
location (mode) and variability of the age-at-death
distribution directly, based on a segmented trans-
formation of the age-at-death distribution (STAD).
The authors modelled and forecasted the differences
between a standard and an observed distribution,
using a transformation function that depends on
the changes in these differences in the mode, differ-
ences in variability before the mode, and differences
in variability after the mode. This model directly cap-
tures the two main mortality dynamics: shifting and
compression of mortality. Oeppen (2008) also devel-
oped a forecasting model based on the age-at-death
distribution. He used compositional data analysis
(CoDA) to model and forecast a redistribution of
28 Marie-Pier Bergeron-Boucher et al.

deaths across age groups (usually from younger to
older ages). Compositions are vectors containing
positive values that represent parts of a whole and
carry relative information summing up to a constant
(e.g. proportions). CoDA is a set of tools that allows
for the correct modelling of compositions, including
distributions (Aitchison 1982). The use of the age-
at-death distribution and CoDA in forecasting
takes age dependency into account: due to the
sum constraints, deaths in the life table are directly
dependent on each other at the aggregate level,
such that a decrease in deaths at one age will
lead to an increase in deaths in at least one other
age group. This property of the model resolves
independence problems between mortality com-
ponents in forecasting, including causes of death
(Kjærgaard et al. 2019). The use of CoDA for fore-
casting mortality has attracted attention in recent
years (Bergeron-Boucher et al. 2017; Kjærgaard
et al. 2019; Kjærgaard et al. 2020; Shang et al.
2022).
In this paper, we aim to develop a new forecasting
model that is based on the regularity of the modal
age at death and that can also account for changes
in lifespan variation. We use a CoDA approach to
forecast the distribution centred around the modal
age at death and forecast M independently. We call
our model the Mode model. It captures both the
shifting and compression of mortality, directly mod-
elling the two main mortality dynamics as two dis-
tinct metrics.
The paper is organized as follows. In the Data
section, we describe the data set used and the
populations analysed. The Methods section starts
with a description of a smoothing procedure, and
we then describe how we estimate and forecast
the modal age at death. We next introduce a
CoDA model for forecasting the age-at-death dis-
tribution centred around the modal age at death;
this model captures changes in lifespan variation.
The remaining Methods subsections present how
we calculate prediction intervals and how we esti-
mate forecasting accuracy via an out-of-sample
analysis, respectively. In the Results section, we
provide an illustration of the methods by forecast-
ing mortality for females and males in 10 West
European countries. First, we present the par-
ameters of the model and discuss their interpret-
ation; second, we present the results of the out-
of-sample analysis, comparing the proposed model
with four other models; and, third, we show the
forecasts up to 2050. Finally, in the Discussion,
we review the methods and results, adding conclud-
ing remarks.
Data
We forecast mortality in 10 West European
countries: Denmark, Finland, France, Ireland, the
Netherlands, Norway, Portugal, Spain, Sweden, and
Switzerland. This provides a mixture of countries
with low, medium, and high mortality levels, as well
as faster and slower rates of mortality progress. See
Appendix A in the supplementary material for
further information on the selection of countries.
We use death counts and exposures from the
Human Mortality Database (HMD 2022) from
1960 to 2019 (the last year with available data for
all countries), for males and females aged 40–110
by single year of age.
Methods
We introduce a new forecasting model, labelled the
Mode model, which is based on change in the
modal age at death. The model consists of three
main steps: (1) smoothing; (2) estimating and fore-
casting the modal age at death; (3) estimating and
forecasting the age at death distribution centred
around the mode.
Smoothing
Mortality is smoothed using a Poisson P-spline
approach. We take the observed unsmoothed death
counts and exposure from the HMD and smooth
them using the R package MortalitySmooth
(Camarda 2012). From these smoothed death rates,
we calculate life tables. The smoothed age-at-death
distribution is necessary for estimating the modal
age at death, as described next.
Estimating and forecasting the modal age at
death
The modal age at death (M) is the age at which the
most deaths occur. The estimation of M is not
straightforward, as the age-at-death distribution is
not always smooth and random fluctuations can
result in multiple maxima in the density of adult
deaths. Several approaches to overcoming the ir-
regular patterns of deaths around the mode have
been suggested. Parametric models, such as the
Gompertz or Siler models, have been used to
smooth the mortality curve and calculate the mode
(Canudas-Romo 2008). However, these models
Forecasting with the modal age at death 29

assume specific mortality shapes. Non-parametric
approaches for estimating M have also been
suggested, for example the Kannisto method
(Kannisto 2001) and the P-spline approach
(Ouellette and Bourbeau 2011). In this paper, we
follow the method for estimating M suggested by
Ouellette and Bourbeau (2011), where the age-at-
death distribution is smoothed with a Poisson
P-spline approach (Eilers and Marx 1996).
M has been increasing linearly since the middle of
the twentieth century in most developed countries
(Horiuchi et al. 2013). As a result of this linear devel-
opment, we forecast M using a random walk with
drift.
Estimating and forecasting the distribution
around M
The forecast of the distribution around M requires
three sub-steps: (1) recentring the age-at-death dis-
tribution around M; (2) forecasting the distribution
with a CoDA model; and (3) completing the distri-
bution for ages not supported by the forecasted
distribution of deaths.
First, the age-at-death distribution around M can
be expressed as a vector, d=[dM−20,dM−19 ,...,
dM−1,dM,dM+1,...,dM+20], with dx being the life-
table deaths. This vector can be interpreted as an
indicator of lifespan variation, that is, how com-
pressed the lifespan distribution is around M. The
dx centred around M are arranged in a matrix by
time t and age x−M, which is notated D, where
each row sums to unity and thus is compositional
data. The location (M) and scale (D) of age-at-
death distributions are then forecasted separately,
that is, we model and forecast the shifting and
compression of mortality as two separate metrics.
Figure 1 illustrates changes in M and D over time
for males in France. Figure 1(a) shows the linear
development of M, and Figure 1(b) shows that
deaths are increasingly concentrated around M
over time.
Second, the smoothed age-at-death distribution
around M can be forecasted using the CoDA
approach (Oeppen 2008). In the case of mortality
compression, the model forecasts a redistribution
of deaths towards M and is defined as:
clr(dt,x−M⊖
a
x−M)=
k
t
b
x−M+1t,x−M, (1)
where dt,x−M is the age-at-death distribution at time
t and age x−M, and
a
x−M is the age-specific
average of dt,x−M over time. The clr is the
centred log-ratio transformation, and ⊖is a pertur-
bation operator (Aitchison 1982). The parameter
k
t
is the time index, and
b
x−M is the age-specific sen-
sitivity to
k
t. The latter indicates which ages gain
deaths over time, relative to M, and which lose
deaths in relative terms. The parameters
k
t and
b
x−M are estimated from a generalized singular
value decomposition (GSVD). The GSVD allows
weights to be assigned to the age and time dimen-
sions. We use a similar approach to that of
Kjærgaard et al. (2019) to assign the weights. The
age-specific weights are determined by the mean
age-at-death distribution over time, thus giving
more weight to the mode and the ages around it
than to ages further from the mode. For the
weights on the time dimension (wt), we follow
the approach of Hyndman et al. (2013), where
more weight is given to the latest years observed
than to earlier years:
wt=
r
(1 −
r
)T−t, (2)
where
r
determines the percentage weight for the
most recent year (T). We use a
r
of 5 per cent
as suggested by Kjærgaard et al. (2019).
The time index parameter,
k
t, is not always linear.
This feature is also observed when looking at
measures of lifespan variation (Edwards and
Tuljapurkar 2005; van Raalte et al. 2018). When fore-
casting non-linear trends, Hyndman and Athanaso-
poulos (2018) suggested using a natural cubic
smoothing spline (which is a cubic spline with some
constraints), so that the spline function is linear at
the end (Hyndman et al. 2012). We use this approach
to forecast
k
t.
Third, as M increases, the age range that sup-
ports the distribution of deaths in the forecast
varies over time. More precisely, we obtain less
and less information about the left tail of the distri-
bution, that is, mortality at young ages. To remedy
this problem, we assume that the missing values in
the left tail of the distribution are equal to
dt,x+1×􏽣
Rx, where 􏽣
Rx is the ratio of dt,x between
two consecutive ages at the last year of obser-
vation: 􏽣
Rx=dT,x
dT,x+1. We also tested extrapolating
the left tail using the penalized composite link
model for ungrouping (Rizzi et al. 2015) and a
monotonic interpolating spline and find that the
model based on ratios provides the most satisfac-
tory results. As dt,x values are usually small at the
beginning of the selected age interval, how we
30 Marie-Pier Bergeron-Boucher et al.

estimate the missing dt,x has only a minor impact
on the forecast results.
Prediction intervals
Prediction intervals are estimated by a bootstrapping
procedure based on fitting errors from both time-
series models used to forecast M and
k
t. For each
time series (M and
k
t), we compute 200 simulations.
For each simulation of
k
t, we calculate the correspond-
ing D. We then calculate age-at-death distributions
for all possible combinations of simulated M and D,
corresponding to 40,000 (200 × 200) simulated distri-
butions. Relevant indicators (e.g. life expectancy and
lifespan variation) are calculated for all simulations,
and 95 per cent prediction intervals are calculated
from the simulations by taking the 2.5th and 97.5th
percentiles. We do not account for the errors
induced by smoothing the age-at-death distribution
when calculating the prediction intervals.
Out-of-sample approach
We use an out-of-sample approach based on differ-
ent fitting periods and forecast horizons to evaluate
forecasting accuracy. Forecasts are sensitive to both
these components. We forecast mortality for each
country starting from each year between 1994 and
2014 and forecasting up to 2019, representing fore-
cast horizons varying from 25 to five years (21 fore-
casts). For each forecast horizon, the fitting period
starts in either 1960, 1965, 1970, or 1975 (four
forecasts) and thus its length varies from 55 to 20
years. As a result, we make 84 forecasts (21 × 4) for
each country and sex.
We measure accuracy by the root mean square
error, using different indicators: (1) life expectancy
at age 40, e40; (2) the modal age at death, M; (3) life-
span variation; and (4) logarithmic age-specific death
rates, ln(mx). Lifespan variation is measured by the
average years of life lost at age 40, e†(Vaupel and
Canudas-Romo 2003). The errors in the logged
death rates are more important at younger ages rela-
tive to older ages, due to the higher logged values at
younger ages. As such, mean errors in forecasting the
age-specific logged death rates are driven by errors
at younger ages, even if mortality at these ages is
low. To remedy this problem, we weight the errors
in logged death rates by the observed age-at-death
distribution. Finally, the accuracy of the prediction
intervals for life expectancy at age 40 is assessed by
the percentage of observed values falling within the
intervals.
We compare the accuracy of the Mode model with
that of the LC model (Lee and Carter 1992), the
functional data approach (FDA) (Hyndman and
Ullah 2007; Hyndman et al. 2012), the CoDA
model (Oeppen 2008), and the STAD model (Base-
llini and Camarda 2019). We select the LC model as
this model has established itself as the standard and
is one of the most used forecasting models (Basellini
et al. 2023). We select the FDA as one possible
variant of the LC model which generally improves
its accuracy (Booth et al. 2006; Hyndman and
Ullah 2007; Hyndman et al. 2012). The CoDA
model is selected because our Mode model uses
Figure 1 (a) Modal age at death, M, and (b) age-at-death distribution around the mode, D: males in France,
1965–2019
Source: Authors’ analysis of data from HMD (2022).
Forecasting with the modal age at death 31

CoDA to forecast the variance around the mode and
this comparison can show evidence of whether using
the mode can improve the model, based on a similar
method. Finally, the STAD model is selected as it is,
to our knowledge, the only other model using the
mode as an input to forecast period life expectancy.
However, the modal age at death has been used as
an input to forecast cohort mortality in the model
of Rizzi et al. (2021). Predictions intervals for all
selected models are based on a bootstrapping pro-
cedure on the time series, as suggested previously
by Bergeron-Boucher et al. (2017) for the CoDA
model and by Basellini and Camarda (2019) for the
STAD model. The traditional LC model uses the
innovation to calculate prediction intervals, which
tends to lead to intervals that are too narrow. But
methods using the bootstrap procedure have been
suggested for the LC model, and these tend to lead
to wider prediction intervals (Keilman and Pham
2006). A jump-off correction is carried out for all
compared models, so that the observed and fitted
values of the input (death rates or age-at-death dis-
tribution) at the jump-off year are the same. Correct-
ing for the error at jump-off year improves accuracy
for all models compared.
Results
Parameters
Figure 2 shows the estimated parameters M,
k
t, and
b
x−M, and their forecasted values for males and
females in France. M has been increasing linearly in
France since the 1970s for males and since before the
1960s for females (Figure 2(a)). Similar results (not
shown) are also found for the other nine countries,
with the increase starting in the 1990s at the latest, for
males in Denmark. Thus, a postponement in the mor-
tality schedule has been seen in the last three to six
decades across all countries and both sexes. Trends in
the mode have generally been linear, which makes
them easily extrapolatable by a linear model.
The parameters
k
t and
b
x−M describe the change
in the age-at-death distribution around M and
capture how lifespan variation evolves over time.
When
k
t increases (Figure 2(b)), deaths are redistrib-
uted from ages with negative
b
x−M towards ages with
positive
b
x−M. Figure 2(c) shows that deaths have
become increasingly redistributed towards M and
the ages around it over time, capturing a com-
pression of mortality around M. The
b
x−M profile
is very similar across countries for males, but more
variations are observed for females (see Appendix
B, supplementary material). However, for all the
selected countries and for both sexes, there has
been a redistribution of deaths from ages about 10
years and more above the mode towards the modal
age at death (highest value) and ages around it.
This result indicates a compression of mortality
around the increasing mode.
Out-of-sample analysis
Table 1 shows the mean forecasting errors in life
expectancy at age 40 (e40), the accuracy of the predic-
tion intervals for e40, and the mean forecasting errors
in modal age at death (M), lifespan variation (e†),
and weighted logged age-specific death rates
(ln(mx)) for five models: Mode, STAD, CoDA,
FDA, and LC. Accuracy levels vary by sex,
country, and model. However, on average, the
Mode model is the most accurate in predicting e40,
M, and ln (mx) for both males and females (see
final column). As mortality postponement has been
the main driver behind the increase in life expect-
ancy since the middle of the twentieth century
(Bergeron-Boucher et al. 2015), it should not be sur-
prising that the model which can best predict M can
also best predict life expectancy.
Bohk-Ewald et al. (2017) have noted that the
evaluation of forecasting models based on life
expectancy alone is not sufficient, as such an evalu-
ation cannot determine whether or not the underlying
mortality developments are plausible. They suggest
also evaluating whether lifespan variation forecasts
are plausible, to check that forecasts can accurately
predict the mortality scale. As shown in Table 1, fore-
casting errors for e†are generally smaller than those
for M and e40 for all selected models. As a result, the
accuracy of the models is not undermined by looking
at this indicator. Although the STAD model’s forecast
accuracy is fair in predicting M (for males), it provides
a relatively low accuracy for e†. The Mode model is
more accurate than the STAD model in forecasting
e†, but the LC and FDA models are the models
which provide, on average, the best accuracy for
lifespan variation.
The CoDA model provides the most accurate
prediction intervals for males, with 90.9 per cent of
the observed life expectancy values falling within
the prediction’s bands, on average. For females, the
FDA model shows the most accurate prediction
intervals, on average, with 95.1 per cent of the
observed values falling within the interval. Using
the Mode model yields 87.3 and 97.8 per cent accu-
racy for males and females, respectively.
32 Marie-Pier Bergeron-Boucher et al.

The Mode model is the most accurate in forecast-
ing age-specific death rates for both sexes, followed
by the CoDA model for males and the FDA model
for females. The CoDA model forecasts an increase
in the ASRMI over time (Bergeron-Boucher et al.
2017), capturing more accurately, compared with
other models, the recent accelerating decline in
mortality at older ages for males and, to some
extent, for females (ranked third for females). In
Appendix C, supplementary material, we show that
the ASRMI at older ages also increase with the
Mode model, but eventually level off, unlike in the
CoDA model.
Forecasts
Figure 3 shows the life expectancy at age 40, modal
age at death, and lifespan variation, observed from
1965 to 2019 and forecasted up to 2050 with the
Mode and LC models, for males and females in
France. The Mode model forecasts that life expect-
ancy at age 40 in France will increase from 41.2
years in 2019 (between 38.9 and 42.9 years for the
other countries) to 47.6 years (44.9–49.3 for the
other countries) in 2050 for males and from 46.5
(44.1–46.8) years to 51.8 (48.4–51.9) years for
females (Figure 3(a)). In comparison, the LC
model forecasts an increase to 45.5 years (between
43.6 and 47.7 years for the other countries) by 2050
for males and 50.5 (46.2–51.0) for females. Com-
pared with the LC model, the Mode model forecasts
a faster increase in life expectancy and modal age at
death (Figure 3(a–b)) but a slower decrease in life-
span variation (Figure 3(c)). The LC model tends
to produce a change in the trends at the jump-off
year, particularly for the modal age at death. The
LC generally assumes that future gains in life
Figure 2 Observed and forecasted model parameters for the (a) modal age at death, M; (b) time index,
k
t; and
(c) age pattern,
b
x−M: males and females in France, 1965–2050
Source: As for Figure 1.
Forecasting with the modal age at death 33

Table 1 Mean forecasting errors across 84 out-of-sample forecasts for life expectancy at age 40 (e
40
), prediction interval accuracy for life expectancy at age 40, and mean forecasting
errors in modal age at death (M), lifespan variation (e
†
), and weighted logged death rates (In(m
x
)), from the Mode, STAD, CoDA, FDA, and LC models: males and females in 10
West European countries
Males
CHE DNK ESP FIN FRA IRL NLD NOR PRT SWE Mean (rank)
e40
Mode 0.41 1.57 0.54 0.31 0.42 1.18 1.31 1.08 0.60 0.48 0.79 (1)
STAD 1.42 1.74 0.82 0.42 1.62 1.37 1.31 1.34 0.68 0.73 1.15 (4)
CoDA 0.75 1.74 0.52 0.54 0.45 1.98 1.68 1.44 1.24 0.96 1.13 (3)
FDA 0.92 1.59 1.62 0.59 1.00 1.46 1.80 1.11 1.28 1.05 1.24 (5)
LC 1.17 1.59 0.66 0.60 0.92 1.33 1.38 1.28 0.99 1.27 1.12 (2)
95 per cent prediction interval
Mode 99.2 93.3 99.9 100.0 98.8 98.068.6 83.8 99.9 99.087.3 (2)
STAD 52.9 55.1 69.9 96.6 56.9 69.1 60.0 57.2 80.0 84.6 68.2 (5)
CoDA 98.4 80.1 97.8 99.9 99.8 86.4 67.481.8 98.4 99.0 90.9 (1)
FDA 90.7 67.8 76.7 100.0 82.1 73.6 51.3 81.999.7 85.4 80.9 (3)
LC 89.2 39.2 94.7 93.8 96.164.7 60.9 49.9 94.788.3 77.1 (4)
M
Mode 0.42 2.23 0.71 0.79 0.34 1.96 1.77 1.43 0.90 0.61 1.12 (1)
STAD 1.53 2.93 1.17 1.12 1.59 2.56 2.03 1.95 1.32 1.11 1.73 (2)
CoDA 1.43 3.02 1.10 1.64 0.78 3.13 2.87 2.40 2.09 1.72 2.01 (4)
FDA 1.38 2.85 2.41 1.73 1.63 3.09 3.03 2.10 2.16 1.66 2.20 (5)
LC 1.73 2.80 1.22 1.76 1.51 2.86 2.45 2.07 1.65 1.84 1.99 (3)
e†Mode 0.19 0.17 0.29 0.29 0.23 0.18 0.12 0.19 0.11 0.10 0.19 (4)
STAD 0.38 0.37 0.51 0.36 0.39 0.17 0.11 0.16 0.18 0.16 0.28 (5)
CoDA 0.16 0.32 0.13 0.17 0.14 0.14 0.18 0.17 0.08 0.17 0.17 (2)
FDA 0.16 0.21 0.24 0.18 0.12 0.29 0.17 0.21 0.08 0.14 0.18 (3)
LC 0.19 0.21 0.15 0.22 0.10 0.22 0.08 0.12 0.08 0.19 0.16 (1)
ln (mx)
Mode 0.86 1.86 0.82 1.07 0.78 1.75 1.66 1.60 0.98 0.84 1.22 (1)
STAD 2.08 2.15 1.30 1.52 2.02 2.34 1.85 1.85 1.29 1.27 1.77 (5)
CoDA 1.05 1.78 0.78 1.09 0.75 2.27 1.73 1.66 1.46 1.15 1.37 (2)
FDA 1.24 1.85 1.65 1.24 1.13 2.19 1.88 1.62 1.55 1.25 1.56 (4)
LC 1.53 1.82 0.97 1.36 1.14 2.14 1.61 1.72 1.42 1.46 1.52 (3)
34 Marie-Pier Bergeron-Boucher et al.

Females
CHE DNK ESP FIN FRA IRL NLD NOR PRT SWE Mean (rank)
e40
Mode 0.53 1.06 0.50 0.40 0.60 0.65 0.37 0.36 0.50 0.29 0.53 (1)
STAD 0.43 1.30 0.78 0.40 0.65 1.49 0.42 1.40 1.22 0.19 0.83 (5)
CoDA 1.11 1.34 0.45 0.68 0.85 1.22 0.53 0.71 1.03 0.28 0.82 (4)
FDA 0.31 1.44 0.38 0.29 0.39 1.02 0.65 1.03 0.56 0.17 0.62 (2)
LC 1.21 1.38 0.33 0.31 0.41 0.89 0.54 0.57 1.67 0.20 0.75 (3)
95 per cent prediction interval
Mode 99.0 91.5 95.2100.0 92.9 99.8 99.8 100.0 100.0 99.7 97.8 (2)
STAD 99.9 65.0 96.5 99.899.3 92.891.6 48.1 93.3 99.788.6 (5)
CoDA 98.9 99.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.8 (4)
FDA 98.8 79.4 100.0 100.0 98.2 83.1 93.8 97.899.8 100.0 95.1 (1)
LC 98.262.2 99.3 99.9 96.578.4 93.7 99.3 93.3100.0 92.1 (3)
M
Mode 0.68 0.50 0.52 0.45 0.52 1.01 0.38 0.22 0.51 0.44 0.52 (1)
STAD 0.46 0.86 0.77 0.46 0.38 2.32 0.54 1.61 1.02 0.36 0.88 (5)
CoDA 0.98 0.81 0.34 0.54 0.39 1.65 0.53 0.78 1.08 0.22 0.73 (4)
FDA 0.25 0.78 0.45 0.47 0.41 1.14 0.64 1.02 0.57 0.19 0.59 (2)
LC 1.05 0.71 0.38 0.42 0.32 1.12 0.49 0.54 1.48 0.21 0.67 (3)
e†Mode 0.27 0.50 0.10 0.15 0.09 0.15 0.22 0.19 0.22 0.15 0.20 (3)
STAD 0.18 0.57 0.16 0.15 0.13 0.11 0.28 0.29 0.23 0.21 0.23 (4)
CoDA 0.27 0.51 0.33 0.19 0.33 0.16 0.15 0.17 0.17 0.08 0.24 (5)
FDA 0.10 0.54 0.18 0.13 0.15 0.12 0.19 0.22 0.10 0.08 0.18 (1)
LC 0.25 0.45 0.21 0.15 0.15 0.12 0.13 0.12 0.21 0.08 0.19 (2)
ln (mx)
Mode 1.08 1.31 0.82 1.04 0.87 1.24 0.75 0.91 0.93 0.74 0.96 (1)
STAD 1.16 1.59 1.10 1.41 0.92 2.34 0.94 2.09 1.73 0.88 1.42 (5)
CoDA 1.37 1.46 0.88 1.20 1.10 1.64 0.80 1.09 1.31 0.68 1.15 (3)
FDA 0.91 1.59 0.86 1.16 0.78 1.65 0.96 1.48 1.00 0.73 1.11 (2)
LC 1.48 1.59 0.85 1.30 0.79 1.69 0.92 1.23 1.92 0.82 1.26 (4)
Note: Values in bold indicate the model providing the most accurate forecasts. Mean forecasting errors are across the 84 out-of-sample forecasts for each country. The final column shows the mean errors
across all countries combined.
Source: Authors’ analysis of data from HMD (2022).
Forecasting with the modal age at death 35

expectancy will result from faster compression but
slower postponement than gains in the past. As the
increase in the mode is due mainly to mortality
reduction above it, this result might come from the
low and constant ASRMI at older ages assumed by
the LC model. Meanwhile, the Mode model allows
for increasing ASRMI at older ages. The model gen-
erally produces increasing ASRMI above age 85,
constant or decreasing ASRMI between ages 65
and 85, and mixed trends below age 65, depending
on the country and sex (see Appendix C, supplemen-
tary material).
The forecasts using the Mode model generally
continue past trends in e40 , M, and e†, without
noticeable trend breaks. Figure 4(a) shows that the
forecasted trends in life expectancy at age 40 stay
somewhat consistent for individual countries, but a
divergence in trends is forecast between countries,
with an increase in the range of life expectancy
from 3.0 years in 2019 to 4.4 years in 2050 for
males and from 2.7 to 3.5 years for females.
For females, M looks similar across all countries,
except for Finland, Ireland, and Portugal, until the
early 1990s. After then, the trends started to
diverge, with slower improvements for Denmark,
the Netherlands, and Sweden, where M reached
lower levels similar to those in Finland, Ireland,
and Portugal. The forecasts show persisting future
divergence. For the Netherlands and Sweden,
trends in e40 and M are consistent, which might indi-
cate that mortality at older ages in these countries is
not decreasing as rapidly as in other countries. For
Figure 3 (a) Life expectancy at age 40, e40; (b) modal age at death, M; and (c) lifespan variation, e†, observed
and forecasted with the Mode and Lee–Carter models: males and females in France, 1965–2050
Note: The shaded areas represent the 95 per cent prediction intervals.
Source: As for Figure 1.
36 Marie-Pier Bergeron-Boucher et al.

females in Denmark, although e40 stagnated between
1975 and 1995, we do not observe such great stagna-
tion in M, despite slower improvement after 1990.
This result hints that the e40 stagnation for
Denmark resulted from mortality worsening below
the mode, as shown by the increase in e†(Figure 4
(c)) and by other studies (Christensen et al. 2010;
Lindahl-Jacobsen et al. 2016).
For males, the divergence in e40 and M can be
explained by the use of 1965–2019 as the fitting
period for all countries. The mode started to increase
only in the 1980s for males in the Netherlands and
Sweden and in the 1990s for males in Denmark,
but it has been increasing since 1970 for France,
Switzerland, and Portugal. The forecasted increases
in M may thus be slowed down by fitting the model
over a period of stagnation, an issue most important
in Denmark, the Netherlands, and Sweden.
Some divergence is also forecast in lifespan vari-
ation, mainly due to the influence of males in Portu-
gal and females in Denmark (Figure 4(c)). The range
of values of e†increases from 1.3 years in 2019 to 1.6
years in 2050 (1.3 without Portugal) for males and
from 0.8 to 1.0 years for females (0.5 without
Denmark). Since the 1990s there has been a small
improvement in e†for males in Portugal and a
rapid decrease for females in Denmark. The Mode
model continues these trends in the forecast, which
Figure 4 (a) Life expectancy at age 40, e40; (b) modal age at death, M; and (c) lifespan variation, e†, observed
and forecasted with the Mode model: males and females in 10 West European countries, 1965–2050
Note: This figure is best viewed online in colour.
Source: As for Figure 1.
Forecasting with the modal age at death 37

leads to a divergence in trends for females in
Denmark and males in Portugal relative to their
counterparts in other countries.
Discussion
The age-at-death distribution provides important
information about mortality patterns and changes,
longevity (postponement), and lifespan variation
(compression). In addition, forecasting with the
age-at-death distribution solves the dependency prob-
lems between components (with the use of specific
models) and provides less biased forecasts than
similar models based on death rates (Bergeron-
Boucher et al. 2017; Kjærgaard et al. 2019). Yet,
age-at-death distributions are not commonly used
as inputs for forecasting models, especially com-
pared with age-specific death rates or probabilities.
In this paper, we developed a model that builds on
at least three major advantages that result from con-
sidering the age-at-death distribution as an input: the
Mode model (1) uses the important regularity of an
almost linear change in the modal age at death; (2)
accounts for changes in lifespan variation; and also
(3) considers the dependence between ages due to
its use of the CoDA model. Our analysis revealed
that the Mode model could, on average, better
predict both the modal age at death and life expect-
ancy at age 40 in 10 West European countries and for
both sexes, compared with the other models con-
sidered. It also provides more accurate forecasts of
the age-specific death rates and plausible trends in
lifespan variation.
The modal age at death has increased linearly
since the second half of the twentieth century in
many low-mortality populations (Horiuchi et al.
2013), providing a strong basis from which to extrapo-
late past trends. In contrast to life expectancy, the
modal age at death is not sensitive to potential
increases or slowdowns in mortality at ages below
it (Canudas-Romo 2010). A stagnation or decrease
in life expectancy has been observed over some
periods of time in different countries, including
Denmark and the United States (US). It is often
caused by mortality worsening at young- or mid-
adult ages (Lindahl-Jacobsen et al. 2016; Woolf and
Schoomaker 2019). For example, in Denmark, life
expectancy stagnated for females between 1975
and 1995, due mainly to limited mortality improve-
ment at mid-adult ages and a considerable burden
from cancer mortality in specific birth cohorts
(Christensen et al. 2010; Bergeron-Boucher et al.
2019b). However, no such stagnation was observed
at older ages and, as a result, the modal age at
death has been increasing in Denmark since the
1960s (or earlier) for females and the 1990s for
males. Models that take advantage of this regular be-
haviour of the modal age at death can produce more
accurate mortality forecasts for this country. The
modal age at death is usually better suited than life
expectancy to capturing the location of the age-at-
death distribution, the speed of its shifting, and the
increase in longevity.
Measures of central tendency, such as the mode or
mean, are not sufficient to determine whether mor-
tality forecasts are plausible, as similar values of
the mode or mean can result from different mortality
developments (Bohk-Ewald et al. 2017). Lifespan
variation provides useful information about mor-
tality scales and inequalities. A reduction in lifespan
variation is observed in most populations, with
deaths becoming increasingly compressed around
the modal age at death (Kannisto 2001). The Mode
model directly models and forecasts this dynamic,
with the
b
x−M parameter capturing this redistribu-
tion of deaths towards the mode. Our results
showed that the model can provide plausible trends
in lifespan variation and good forecasting accuracy.
Another advantage of the Mode model is that its
ASRMI can change over time, increasing at some
ages and decreasing at others. Evidence shows that
in low-mortality countries, mortality decline has
been decelerating at younger ages but accelerating
at older ages (Rau et al. 2008; Li et al. 2013;
Vaupel et al. 2021). This pattern is referred to as a
rotation (Li et al. 2013), which the Mode model is
able to capture. The Mode and CoDA models
allow for increasing ASRMI at older ages and gener-
ally produce more accurate forecasts of old-age mor-
tality. At younger ages, the ASRMI tend to stagnate
or slow down. The Mode model is also able to
account for this dynamic. For ages below the mode,
assuming constant or decreasing ASRMI (as implicit
in the Mode and LC models) leads to better forecasts
of the age-specific death rates.
Like other extrapolative models, the Mode model
is sensitive to the fitting period selected. How to find
the most relevant fitting period remains an open
question. Generally, a longer fitting period will
provide more accurate forecasts, but it is also impor-
tant to select a fitting period that reflects an ongoing
or emerging dynamic (Janssen and Kunst 2007; Ber-
geron-Boucher et al. 2019b). The year when M
started to increase—a way of capturing the mortality
postponement dynamic—differed by country and
sex. The use of the same fitting period across all
studied populations, for consistency, affected the
38 Marie-Pier Bergeron-Boucher et al.

accuracy of the forecasts. For males, using a
more recent fitting period might have been more
suitable.
Our proposed model is heavier in terms of its
number of steps than other models, such as the LC
model that directly forecasts age-specific mortality
rates. Simplicity can help a model to be more
widely used and is often cited as one of the main
advantages of the LC model (Basellini et al. 2023).
But sacrificing some simplicity can be justified if
forecast accuracy can be significantly improved.
The literature shows multiple examples of reason-
able multistep approaches in forecasting. For
example, the STAD model forecasts the com-
pression and shifting dynamics independently. The
United Nations (UN) forecasts life expectancy and
then derives age-specific mortality patterns with a
separate model (Raftery et al. 2012). Coherent fore-
casting models usually forecast a standard independ-
ently and forecast the population-specific
deviation from this average as two separate steps;
this generally improves the forecasts (Li and Lee
2005; Booth 2020).
Due to the Covid-19 pandemic, life expectancy
declined in most countries in 2020 and 2021
(Aburto et al. 2022). Extrapolative models, such as
those tested in this paper, cannot account for mor-
tality shocks. As the last year observed for all
selected countries in the HMD was 2019, the pre-
dicted life expectancies for 2020 and 2021, and
potentially 2022, will most likely be too high. Pre-
vious mortality shocks, such as the Spanish flu and
the two world wars, led to a decrease in life expect-
ancy for a short period of time, after which life
expectancy returned to its prior level within one or
two years (Schöley et al. 2022). In this context, the
models tested should still be relevant to forecasting
post-pandemic mortality. However, whether or
when mortality will return to its expected trajectory
is still unclear.
The Mode model is limited to forecasting adult
mortality patterns. Due to the shape of the human
mortality pattern, using the full age range will
create some fitting problems. For example, there is
a second mortality peak at birth. When estimating
the age-at-death distribution centred around M, the
peak of infant deaths will be allocated to ages
further and further away from M over time, creating
an artificially fast mortality decline at the relative
ages where the peak used to be. A similar issue will
arise if we consider the ages at which the accident
mortality hump appears (around ages 20–30). For
this reason, we suggest limiting the forecast to mor-
tality at ages 40 and above.
The proposed model has not been tested in cases
where lifespan variation increases, such as the US
since the 2010s (Acciai and Firebaugh 2019). It is
unclear how the model will perform in such a
context: this will depend on the
b
x−M pattern. If
b
x−M captures a redistribution of deaths towards
ages below the mode, we might forecast an indefinite
increase in lifespan variation. In such a context, an
extension of the model could be developed to
allow for changes (rotations) in
b
x−M over time, as
previously suggested for the LC model (Li et al.
2013).
The Mode model can also be extended to reflect
other mortality processes. For example, a coherent
version of the model could be developed to
account for non-diverging trends between countries
or sexes, by forecasting the differences between the
modal age at death and a reference trend. The
changes in lifespan variation between countries
could also be forecast by applying the coherent
CoDA model from Bergeron-Boucher et al. (2017).
Cause-of-death information or smoking-related
mortality data could also be included in the model
(Janssen et al. 2013; Kjærgaard et al. 2019). The use
of CoDA makes the Mode model particularly
adept at forecasting mortality by cause, due to its
component-dependence modelling. We forecasted
the modal age at death and the time index of the
age-at-death distribution centred around the mode
as two separate trends because, sometimes, they
behave inconsistently. However, the two measures
are very often negatively correlated and, in such
cases, both indicators can be forecasted dependently.
Kjærgaard et al. (2019) suggested using a co-
integrated vector error model to account for depen-
dence between multiple time indexes. It was,
however, outside the scope of this paper to test all
possible extensions of the proposed model.
Our model captures the two main mortality
dynamics at adult ages: compression and shifting.
These components are also well captured by
the STAD model. The main difference between the
two models is that the STAD model forecasts the
difference between a standard age-at-death distri-
bution and the observed distribution, whereas the
Mode model directly forecasts the observed age-at-
death distribution. The STAD model also forecasts
three sets of parameters (the mode and the variation
before and after the mode), whereas the Mode
model forecasts two sets of parameters (M and
k
t).
We believe that the modal age at death is a solid
basis for forecasting mortality, as this indicator has
been increasing with few or no breaks in many popu-
lations since the second half of the twentieth century.
Forecasting with the modal age at death 39

This regularity, combined with M capturing mortality
postponement and being easily combined with
measures of scale and inequalities, makes the use
of the modal age at death appealing for forecasting
mortality. The Mode model we have introduced pro-
vides several advantages and can improve forecast-
ing accuracy compared with other models. This is,
however, only a first step, and potential extensions
of the model (e.g. its coherent extension) should
help to improve accuracy even more.
Notes and acknowledgements
1 Please direct all correspondence to Marie-Pier Ber-
geron-Boucher, Interdisciplinary Centre on Population
Dynamics, University of Southern Denmark, Campus-
vej 55, Odense 5230, Denmark; or by E-mail:
mpbergeron@sdu.dk.
2 Acknowledgement: This paper is dedicated to Jim
W. Vaupel. He wanted to develop better forecasting
models based on strong regularities in mortality trends
and asked us to investigate how to forecast using the
modal age at death. We are deeply thankful for his crea-
tivity and inspiring discussion. We also want to thank
Silvia Rizzi for her help with ungrouping techniques.
3 Funding: The research leading to this publication is a
part of a project that has received funding from the
ROCKWOOL Foundation, through the research
project ‘Challenges to the Implementation of Indexa-
tion of the Pension Age in Denmark’; from the Euro-
pean Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation pro-
gramme (grant agreement number 884328—Unequal
Lifespans); and by the AXA Research Fund, through
the funding for the AXA Chair in Longevity Research.
Disclosure statement
No potential conflict of interest was reported by the
authors.
ORCID
Marie-Pier Bergeron-Boucher http://orcid.org/
0000-0001-7383-3175
Paola Vázquez-Castillo http://orcid.org/0000-
0001-9680-4721
Trifon I. Missov http://orcid.org/0000-0002-7312-
4261
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