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The aging of population is perhaps the most important problem that developed countries must face in the near future. Dependency can be seen as a consequence of the process of gradual aging. In a health context, this contingency is defined as a lack of autonomy in performing basic activities of daily living that requires the care of another person or significant help. In Europe in general and in Spain in particular, this phenomena represents a problem with economic, political, social and demographic implications. The prevalence of dependency in the population, as well as its intensity and evolution over the course of a person’s life are issues of greatest importance that should be addressed. The aim of this work is the estimation of life expectancy free of dependency (LEFD) based on functional trajectories to enhance the regular estimation of health expectancy. Using information from the Spanish survey EDAD 2008, we estimate the number of years spent free of dependency for disabled people according to gender, dependency degree (moderate, severe, major) and the earlier or later onset of dependency compared to a central trend. The main findings are as follows: first, we show evidence that to estimate LEFD ignoring the information provided by the functional trajectories may lead to non-representative LEFD estimates; second, in general, dependency-free life expectancy is higher for women than for men. However, its intensity is higher in women with later onset on dependency; Third, the loss of autonomy is higher (and more abrupt) in men than in women. Finally, the diversity of patterns observed at later onset of dependency tends to a dependency extreme-pattern in both genders.
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION
OF HEALTH EXPECTANCY: THE CASE OF DISABLED SPANISH
POPULATION
BY
IRENE ALBARRÁN,PABLO J. ALONSO-GONZÁLEZ,ANA ARRIBAS-GIL
AND AUREA GRANÉ
ABSTRACT
The aging of population is perhaps the most important problem that developed
countries must face in the near future. Dependency can be seen as a conse-
quence of the process of gradual aging. In a health context, this contingency is
defined as a lack of autonomy in performing basic activities of daily living that
requires the care of another person or significant help. In Europe in general and
in Spain in particular, this phenomena represents a problem with economic,
political, social and demographic implications. The prevalence of dependency
in the population, as well as its intensity and evolution over the course of a per-
son’s life are issues of greatest importance that should be addressed. The aim
of this work is the estimation of life expectancy free of dependency (LEFD)
based on functional trajectories to enhance the regular estimation of health
expectancy. Using information from the Spanish survey EDAD 2008, we esti-
mate the number of years spent free of dependency for disabled people accord-
ing to gender, dependency degree (moderate, severe, major) and the earlier or
later onset of dependency compared to a central trend. The main findings are
as follows: first, we show evidence that to estimate LEFD ignoring the infor-
mation provided by the functional trajectories may lead to non-representative
LEFD estimates; second, in general, dependency-free life expectancy is higher
for women than for men. However, its intensity is higher in women with later
onset on dependency; Third, the loss of autonomy is higher (and more abrupt)
in men than in women. Finally, the diversity of patterns observed at later onset
of dependency tends to a dependency extreme-pattern in both genders.
KEYWORDS
ADL, cox regression, dependency, disability, functional data.
JEL codes: 62-07, 62-09, 62H20, 62H99, 62P05.
Astin Bulletin 49(1), 57-84. doi:10.1017/asb.2018.34 c
2018 by Astin Bulletin. All rights reserved.
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58 I. ALBARRÁN ET AL.
1. INTRODUCTION
Population aging is an ongoing global phenomenon and a powerful and trans-
forming demographic force. Several reports have warned about the need of
evidence-based policies sustained on rigorous research and on the importance
to prioritize healthy aging and well-being (see for instance, WHO, 2011a,b;
Lloyd-Sherlock et al.,2012). In particular, one of the eight tackling societal
challenges of the European program Horizon 2020 is concerned with these
issues.
The decreasing mortality and increasing life expectancy in most Western
European countries during the last decades are well documented (Eurostat,
2009). A key issue is to find out whether the increased life expectancy is healthy
or either associated with an increase or decrease in disability (Fries, 1983).
Life expectancy is one of the most used indicators to measure quantity of
life. However, if the aim is to measure quality of life, indicators related to
healthy life expectancy should be used. These kinds of indicators introduce
health status (morbility or disability) of the individual (Robine and Ritche,
1991; Robine et al.,2003). For instance, Sanderson and Scherbov (2010)pro-
pose disability-free life expectancy as a consistent disability aging measure for
many countries in order to provide better tools for policy makers. In this paper
we are interested in dependency,1which is a more restrictive concept than
disability. Therefore, our indicator will be life expectancy free of dependency
(LEFD) (see Martel and Bélanger, 2000).
The aim of this work is to estimate the LEFD, that is, the expected number
of years that a person can live free of this contingency based on mortal-
ity and morbility conditions. The evolution of dependency in the disabled
Spanish population will be studied through a pseudo panel constructed from
EDAD 2008, in the lack of longitudinal studies or the possibility to link
different cross-sectional surveys.2
EDAD 2008 (Survey on Disability, Personal Autonomy and Dependency
Situations 2008, undertaken by the Spanish Statistical Office—INE) is the most
recent Spanish survey about disability and was the first Spanish survey that
used the internationally accepted measures established by the ‘International
classification of functioning, disability and health’.3It was also the first time
that the survey included information useful for studying the dependency phe-
nomenon, such as the average hours per week of special care received by the
dependent person.
Our main contribution is the estimation of LEFD based on gender, depen-
dency degree (moderate, severe and major) and homogeneous groups of indi-
viduals with similar dependency pattern. The characterization of homogeneous
groups of individuals is obtained through the proximity of the dependency
trajectories (that are derived using the retrospective reported information of
each individual from birth up to 2008, contained in EDAD 2008) to a central
trend within each age–gender group. These central trends are computed via
functional data techniques. To estimate LEFD in all the scenarios considered,
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 59
we use the specification of Cox regression model in terms of the survival
function, having in mind that the event of interest is not ‘survival’ itself but
‘being dependency-free at a given age’. Finally, we estimate the LEFD for
disabled Spanish population within homogenous groups considering gender,
dependency degree and ages from 30 to 100.
A very preliminary attempt to estimate life expectancy can be found in
Albarrán et al. (2014). The present work is a novelty approach to solve the same
problem and, as far as we know, this is the first time that dependency evolution
is used to characterize the individuals in order to enhance the regular estima-
tion of health expectancy. Other recent studies on dependency are Albarrán
et al. (2015) and Albarrán-Lozano et al. (2017), regarding dependent children.
Other authors used Markovian multi-state models to study long-term care
risk (see, for instance, Biessy, 2017; Fong et al.,2017; Levantesi and Menzietti,
2018), where several states such as autonomy, different degrees of dependency
and death must be established. However, this is not the case in database
EDAD 2008 where all surveyed people was alive in 2008. To circumvent
this problem, in this paper we present a novel approach via functional data
techniques.
The main findings are as follows: first, the relative errors of the LEFD cal-
culated using the partition by proximity-groups versus the global LEFD show
evidence that the global LEFD may not be representative of the Spanish pop-
ulation. From economic and demographic points of view, this is a relevant
finding, since the expected dependent population would demand care services
(health care, pensions and other services) that should be covered and related
expenditures should be financed. Second, in general, healthy life expectancy is
higher for women than for men. However, the intensity of dependency is higher
for those women with later onset of dependency. Third, the loss of autonomy
is higher (and more abrupt) for men than for women. Fourth, for people with
the earliest onset of dependency, having less than 50 points (out of 100) in
the dependency rating scale is crucial for living a longer time free of major
dependency. Finally, the loss of autonomy in people with the earliest onset of
dependency tends to a singular extreme-pattern, characterized by few variable
effects on LEFD estimation, whereas for people with the latest onset of depen-
dency the loss of autonomy has a diversity of patterns that can be associated
to a wide range of variable effects on LEFD. These diversities of patterns are
higher in women than in men.
The paper proceeds as follows. Section 2contains the definition of depen-
dency and its graduation according to the Spanish legislation. Also some
information about the Spanish survey EDAD 2008 is presented. Section 3
is devoted to explain the construction of the dependency trajectories from a
pseudo panel from EDAD 2008, a description of the functional data techniques
that we are going to use and the proximity measure that will help to char-
acterize groups with homogeneous dependency trajectories. This is the most
technical section of the paper. In Section 4we propose the methodology to
estimate LEFD and analyse the main results. Finally we conclude in Section 5.
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60 I. ALBARRÁN ET AL.
2. DEPENDENCY SITUATION IN SPAIN:LEGISLATION AND DATA-SET
EDAD 2008
2.1. Spanish legislation on dependency
When talking about dependency two fundamental aspects must be considered.
First, the definition itself. In the Spanish case, article 2 of Act 39/2006, of 14th
December, on the Promotion, Personal Autonomy and Care for Dependent
persons states that dependency is a ‘permanent state in which persons that for
reasons derived from age, illness or disability and linked to the lack or loss of
physical, mental, intellectual or sensorial autonomy require the care of another
person/other people or significant help in order to perform basic activities of
daily living or, in the case of people with mental disabilities or illness, other
support for personal autonomy’.
Second, the assessment of dependency, which is usually solved using spe-
cific dependency rating scales that take into account the disabilities suffered by
the person jointly with their intensity. Royal Decree 504/2007 rules the evalu-
ation of dependency in Spain. The Spanish dependency rating scale goes from
0 to 100 points and it is categorized in four degrees: non-dependant (less than
25 points), I-moderate (greater or equal to 25 but under 50 points), II-severe
(greater or equal to 50 but under 75 points), III-major (greater or equal to 75
points). See Table A1 in the Appendix for more details.
To acknowledge the entitlement to the benefits of the system, a person must
reach at least the moderate degree, that is, at least 25 points are needed to
be considered dependant in Spain. According to the dependency rating scale
value or score reached by an individual, the Spanish legislation establishes a
minimum level of protection, which is defined and financially guaranteed by
the General State Administration.
2.2. EDAD 2008 survey
In order to provide reliable estimates at the national level, the EDAD 2008
survey was performed around the country using sampling. In particular, a
two-stage sampling was performed, stratified and proportional to the size of
the Spanish autonomous regions (with stratified sampling distribution propor-
tional to population size in stratum, within each Spanish province). Therefore,
each individual in EDAD 2008 is associated to a weight reflecting the popu-
lation group that represents. See INE (2010) for more details on the sampling
methodology.
EDAD 2008 gives information about people with disabilities that were
living either at home or in institutions. In the first case, the survey was pre-
pared interviewing 260,000 people who were living in 96,000 different houses,
whereas for institutionalized people, 11,000 people in 800 centres were asked
about their situation. Interviewed people were not only those suffering disabil-
ities, but also their relatives and/or carers. This survey is based on the concept
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 61
TABLE 1
ESTIMATION OF CHILDREN AND ADULT POPULATION WITH DISABILITY LIVING AT HOME:
95% CONFIDENCE INTERVALS FOR NUMBER AND PREVALENCE RATE.
Disabled people (in thousands) and prevalence rate (in %)
Age
(in years) Total Male Female
Under 6 53–67.8 (1.8–2.3%) 30.9–41.9 (2.1–2.8%) 24.4–27.6 (1.1–1.7%)
Between 6 576.8–648.4 (2.3–2.7%) 316–374.2 (2.5–3.0%) 240.5–285.5 (2.0–2.4% )
and 44
Between 45 897.7–1005.9 (8.0–9.0%) 379.4–437.6 (6.9–8.0%) 505.2–580.4 (8.9–10.3%)
and 64
Between 65 1138–1264.6 (20.8–23.1%) 422.4–487.2 (17.1–19.8%) 703.5–789.5 (23.4–26.3%)
and 79
80 or more 971.8–1079.8 (45.8–50.9%) 277.6–326.2 (36.6–43.0%) 683.1–764.7 (50.0–56.0%)
Total 3740.4–3955.2 (8.2–8.6%) 1488.9–1605.5 (6.6–7.1%) 2226.6–2374.6 (9.3–10.2%)
Source: INE elaboration. Results derived from the weighted survey data.
of self-perceived disability, in accordance with the recommendations of the
World Health Organization. So, the target people is identified through a set
of questions about the possible difficulties they can find in doing some specific
activities. Despite its drawbacks, the main advantage of this strategy is that it
is focused on the daily activities of the individuals and the problems they may
have while doing them, with no consideration of medical matters.
In 2008 the Spanish population ascends to 46.66 million people (23.10
million men and 23.56 million women). According to EDAD 2008, there are
more than 4.1 million Spanish people suffering at least one kind of disability.
Although the global prevalence rate is situated between 8.2% and 8.6% with a
95% of confidence, in the case of people living at home, this rate is lower than
that for people living in institutions (8.4% and 17.7%, respectively). Disability is
related to two main factors: gender and age; until 45 years old, the male preva-
lence is statistically significant greater than the female one. After that age, the
relative incidence is greater for women. In general terms, more than 57% people
with this problem are at least 65 years old, being most of them women. Table 1
contains an estimation (derived from the weighted survey data) of children and
adult population with disability living at home.
The sample selected for the present study is formed by 7446 individuals
and represents 2.35% of the Spanish population in 2008, that is, more than
one million people (325,253 men and 773,079 women). We remind that each
individual in the sample has a weight reflecting the population group that rep-
resents. These weights have been taken into account in all the computations of
this paper. We give more details about the selected sample in Section 4.
The data set obtained from EDAD 2008 contains, among many other vari-
ables, the ages at which each person in the sample has suffered a change in
his/her health condition and his/her current age. Although the survey includes
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62 I. ALBARRÁN ET AL.
the term ‘dependence’ in its title, the questionnaire does not consider any
question on this topic. So, the dependency score is not reported in the sur-
vey but can be computed from the information provided in it and applying
the Spanish legislation (Act 39/2006 and Royal Decree 504/2007). This com-
putation is not straightforward. In particular, the Spanish dependency score
is a sum of the weighted product of several factors that take into account the
disabilities suffered jointly with their severity and the degree of supervision
(average hours per week of special care received). The Spanish legislation clas-
sifies the disabilities into 11 main activities: eating and drinking, control of
physical needs (excretion and urinate), washing, other personal tasks, dress-
ing, maintaining health, mobility, moving inside home, moving outside home,
housekeeping and taking decisions. In turn, each activity contains several tasks.
For example, under mobility activity we find the following tasks: sitting down,
lying down, standing up, changing posture from a siting position and changing
posture from bed. Each activity and task contribute to the dependency score
with different weights according to the age of the individual and the occurrence
(or not) of mental disability or cognitive impairment. See Albarrán and Alonso
(2009) for more details on the computation of the Spanish dependency score.
3. SEARCHING FOR PROXIMITY-GROUPS
One of the objectives of the paper is to search for different patterns within age–
gender groups, that is, we are interested in identifying dependency trajectories
that lie close/far/very far from a central trend of the group. The reason is
that, as we will see in Section 4.2, LEFD can experiment huge variations as
dependency trajectories depart from the central trend of the corresponding
age–gender group. The central trend of each age–gender group will be obtained
by using functional data techniques that we describe in Section 3.2.Westart
by obtaining the dependency trajectories.
3.1. From a pseudo panel to dependency trajectories
The aim of this section is to construct a dependency trajectory for each indi-
vidual in the sample, that is, a curve describing the evolution of the personal
dependency situation over time, and to use functional data techniques to anal-
yse the database. Indeed, in functional data analysis, individual observations
are real functions of time, observed at discrete time points. Each curve pro-
vides the evolution of a certain process for a given individual (see Ramsay and
Silverman, 2005, for an overview). In our case, the process of interest is the
evolution of dependency.
Notice that even if the available data come from a one-time survey, individ-
uals were asked about their whole medical history, so we have information
concerning their dependency situation/score from birth up to 2008. Then,
for the ith individual we observe (ti1,yi1), ...,(tini,yini), respectively, the ages
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 63
FIGURE 1: Examples of dependency curves from birth up to 2008. Dependency score is represented in the
vertical axis and age in the horizontal one.
when changes occur and the dependency scores at these ages, and ai,the
current age (at 2008). From these data, in order to stress the step character
of these curves, we add a first point (0, 0) (only if ti1>0), intermediate points
(tih δ,yih1) between (tih1,yih1)and(tih,yih ), where δis a chosen short period
of time, and a final point (ai,yini) (only if tini<ai). Indeed, even if the person’s
health/dependency condition can be seen as a smooth process, the dependency
score function is piecewise constant since changes in score only take place once
some particular disability status has been reached and recognized according
to the Spanish legislation. These transformed sequences will make up our set
of observations from now on. For the sake of simplicity, we will still refer
to them as (tih,yih )h=1,...,ni,i=1, ...,n. Thus, we have ndiscretely observed
curves y1,...,yndefined in different time intervals [0, ai], i=1, ...,n. We illus-
trate the step character of these curves in Figure 1. Notice that the curves are
non-decreasing, meaning that recovery is not possible (in spite of adaptation
strategies). In order to better interpret this figure, let us focus our attention on
a particular trajectory, for example, the purple one. For this particular case,
we can see three jumps in the score, taking place at ages 28, 36 and 67, which
means that, this particular individual became dependent at the age of 28 with
a dependency score of 31 points (Degree I); at the age of 36 another disability
appeared increasing the score up to 45 points (Degree I); finally, at the age of
67 another disability took place and the score jumped to 56 points (Degree II).
The trajectory also tells us that the current age of this individual at 2008 was
67 years old.
In order to apply any functional data analysis technique, we need func-
tions that are defined over the same interval. One idea would be to consider
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64 I. ALBARRÁN ET AL.
FIGURE 2: Dependency trajectories for all women under study and two selected groups of women.
Dependency score is represented in the vertical axis and age in the horizontal one.
the different cohorts present in the sample and analyse the dependency
trajectories within each cohort. However, this may lead to many different
under-represented cohorts, since the age range of the individuals in the sample
is large. Instead of that, we consider disjoint groups of people in age intervals
of 10 years. Within each age interval [A,A+10) we truncate individual curves
to get them defined in [0, A]. For all the analyses performed in this article, the
first age interval is [50, 60) and the last one is [90, ). Moreover, we are partic-
ularly interested in those people with a dependency score of zero at the age of
30, and from now on, they will be grouped in 10 age–gender intervals (5 groups
per gender) according to their current age at 2008. Notice that we consider
such a group of 30-year-old non-dependent people in order to obtain LEFD
estimates for different dependency scenarios, which are useful for health, eco-
nomic, demographic and insurance contexts. In Figure 2we depict the resulting
dependency trajectories for all women under study and two selected groups of
women with dependency score of zero at the age of 30. The aim of this fig-
ure is to illustrate different patterns of dependency. For example, from panels
(b) and (c) we can see that the density of curves in the last 10 years is greater
for women aged 80 than for women aged 70, meaning that the dependency
situation tends to worsen with age. That is the reason why we have considered
disjoint groups of people according to their current age at 2008, otherwise, due
to the great amount of curves (see panel (a)) it would be not possible to track
this phenomenon.
3.2. Estimating the central trend
Providing a measure of centrality when dealing with functional data is not an
straightforward task. Indeed, not only the levels of the curves matter, but also
their shapes, whose information is more difficult to incorporate to any numer-
ical summary. The problem aggravates if we consider curves for which the
main features are not aligned. It is well known that in this context the sample
pointwise or cross-sectional mean is a poor estimator of the mean behaviour
(Gasser et al.,1984; Kneip and Gasser, 1992; Gasser and Kneip, 1995).
In this context, it is extremely important to use measures of centrality that
can take into account the misalignment between the curves of the sample.
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 65
Indeed, in the particular case of the dependency evolution curves that we study
in this work, it is very natural to consider that the evolution of dependency
may present a common pattern which is accelerated or delayed in some
individuals with respect to others. A general framework for modelling such
trajectories is the so-called time warping model, since it includes any kind of
parametric model in which the individual parameters allow for variations in
scale and phase with respect to some given functional form, such as growth
models, and also semi-parametric models in which this functional form is
unknown and estimated from the data, such as shape-invariant models (see
Wang and Gasser, 1997, for details). Also, notice that we can assume that
observations are free of measurement error since they correspond to the
evaluation, on an official numerical scale, of the particular conditions suffered
by each individual at each moment.
In the time warping model, two approaches to estimate the central trend
or mean behaviour of the data are possible: (i) to align or register the curves
and to compute any desired sample statistic on the registered sample; and (ii)
to define appropriate estimators directly on the observed sample, taking into
account the nature of the data. For the analysis of the dependency data set we
will consider an estimator of the second kind that we describe in the following.
3.2.1. Deepest curve.
The literature on estimators directly defined on the unregistered sample is rel-
atively small. Liu and Müller (2004), Dupuy et al. (2011) or Arribas-Gil and
Romo (2012) are works which are particularly concerned by the definition of
suitable population centrality measures, and their corresponding sample statis-
tics, in the time warping model. However, there might be curves with a typical
shape but taking atypical values (abnormally high or low at some locations)
and, in this case, a registration procedure would neutralize the effect of those
curves with an atypical shape (due to the fact that they may be delayed or accel-
erated with respect to the rest). Therefore, for the analysis of the dependency
data set we will consider the approach of Arribas-Gil and Romo (2012) since it
provides a robust estimator of the central trend for a set of curves.
A way to provide a centrality measure that is robust against the two types
of atypical curves is to use functional depth. Indeed, the deepest curve of a
sample in terms of modified band depth (López-Pintado and Romo, 2009)has
been proven to be an accurate and robust estimator of the central pattern of a
sample of curves in the time warping model (Arribas-Gil and Romo, 2012). It
can be understood as a generalization of the median to functional data because,
intuitively, it is the curve that is most surrounded by other curves. Therefore, it
provides an accurate measure of centrality since: (i) it is a curve geometrically
located in the centre of the sample and (ii) it presents a typical shape because it
is one of the observed curves.
As it was mentioned before, we are interested in estimating the central
trend of each age–gender group. Therefore, for each one of these 10 groups we
compute the deepest curve in terms of modified band depth using the roahd
package in Rby Tarabelloni et al. (2016). As an example, in Figure 3we depict
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66 I. ALBARRÁN ET AL.
FIGURE 3: Dependency trajectories for men and women with their corresponding deepest curves (in bold
red). Dependency score is represented in the vertical axis and age in the horizontal one.
the dependency trajectories with the corresponding deepest curve (in bold red)
for several age–gender groups, where we observe that for people aged 50 the
first score value reached by the deepest curve is lower for women than for men,
meaning that the loss of autonomy is stronger in men than in women at earlier
ages. The contrary happens for later ages, reproducing somehow the behaviour
observed in Table 1.
3.3. Distance to the deepest curve
Once we have estimated the central trend within each age–gender group, we
propose to search for different patterns of trajectories within each group by
computing a proximity measure to the deepest curve.
Therefore, within each group we compute the L2-distance of each trajectory
to the corresponding deepest one multiplied by 1 (or 1) if the trajectory is
most of the time above (or below) the deepest curve. Indeed, if we note yij(·)
the dependency curve of individual iin group j,andmj(·) the deepest curve of
group j, we obtain
dj(i)=
tyij(t)mj(t)2·sign
tyij(t)mj(t), (3.1)
i=1, ...,nj,j=1, ..., 10,
where njis the number of individuals in group j. This yields a numeri-
cal summary for each one of the trajectories that can be used to establish
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 67
FIGURE 4: Histogram for djfor men and women. Vertical lines stand for the quartiles of sets {dj<0}and
{dj0}, respectively. Different colours indicate the degree of departure from dj=0.
different patterns. As we will see later, these patterns will exhibit quite different
dependency-free life expectancies.
In Figure 4we depict the histograms for the proximity measure djdefined
in Equation (3.1) computed over all the trajectories by gender. Notice that the
sign of djindicates whether the trajectory is below or above the deepest curve
within its age–gender group. In particular, negative values of djcorrespond to
trajectories below the deepest curve and, therefore, to individuals with lower
dependency scores than those of the central trend of their age–gender groups
(later onset of dependency compared to the central trend). In fact, the best situ-
ations are expected for the left-tail values of dj. On the other hand, positive val-
ues of djcorrespond to trajectories above the deepest curve and, hence, to indi-
viduals with higher dependency scores than those of the central trend of their
age–gender groups (earlier onset of dependency compared to the central trend).
In this case, the worst situations are expected for large values of dj. With verti-
cal lines and different colours we highlight eight regions corresponding to the
division established by the quartiles of sets {dj<0}and {dj0}, respectively.
Since each set of values {dj<0}and {dj0} has a different meaning,
later/earlier onset of dependency, we propose to compute the LEFD for the
groups of individuals established by the quartiles in each set, yielding to eight
groups for each gender, that we call proximity-groups (see Figure 4where the
regions defined by the quartiles are in different colours). Remind that, we are
also interested in estimating LEFD for the three dependency degrees (moder-
ate, severe, major) within each proximity-group. Depending on the researcher,
other partitions or sets of particular interest, such as extreme observations, can
be considered.
3.4. Summary of the procedure
In the following we briefly describe the main steps to find the proximity-groups
for a given set of dependency trajectories. However, the procedure can be
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68 I. ALBARRÁN ET AL.
applied to track the occurrence and evolution of a certain phenomenon in a
group of individuals, whenever other methodologies cannot be applied.
Remind that, in our case, for each individual we have registered the age
when a disability occurs and the dependency score. We also know their current
ages at 2008.
Step 1 Group the individuals in age–gender groups, since dependency can
be seen as a consequence of gradual aging and it is commonly
assumed that the evolution of this phenomenon is different in men
than in women. In our case, we have considered 10 disjoint groups
(5 per gender) according to their current ages at 2008. In particular,
[50, 60), [60, 70) ..., [90, ).
Step 2 Calculate the deepest curve of each age–gender group. The Rpackage
roahd by Tarabelloni et al. (2016) contains several possibilities. In
our case, we have computed the deepest curve in terms of the modified
band depth (López-Pintado and Romo, 2009).
Step 3 For each individual compute the proximity to the corresponding deep-
est curve using measure djfrom formula (3.1). Remind that we are only
interested in two sets of values per gender: {dj<0}, meaning later onset
of dependency, and {dj0}, earlier onset of dependency.
Step 4 To establish the proximity-groups per gender, use certain quantiles
within sets {dj<0}and {dj0}. In our case, we have used the quartiles,
leading to eight proximity-groups per gender.
4. ESTIMATING LEFD
In this section we propose to use Cox proportional hazards regression model
to estimate the LEFD (Cox, 1972; Cox and Oakes, 1984). The reason is that
we are interested not only in estimating healthy life expectancy, but also in
identifying the variables with persistent, high and low impacts on LEFD. Cox
regression model allows to explore the determinants of life expectancy (or sur-
vival probability) and to estimate hazard ratios of the covariates included in
the model, such as gender, disabilities, etc. (Czado and Rudolph, 2002). This
model is one of the most popular regression techniques in survival analysis and
can be written as follows:
h(t,x)=h0(t) expβx, (4.1)
where xis a vector of explanatory variables, βis the vector of coefficients of
the explanatory variables, function h(t) is the expected hazard at time tand
h0(t) is the baseline hazard function that represents the hazard when all the
explanatory variables are equal to zero.
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 69
In our case, since we are interested in estimating the LEFD, the first step is
to consider the specification of Cox regression model in terms of the survival
function S(t). Using the well-known relationship
S(t)=exp t
0
h(t)dt,
and modelling the hazard function by Equation (4.1), we have that
S(t,x)=exp t
0
h0(t) expβxdt=exp t
0
h0(t)dtexpβx
that can be written as follows:
S(t,x)=S0(t)expβx,
where
S0(t)=exp t
0
h0(t)dt
is the baseline survival function at time t, and, as before, xis the vector of
explanatory variables and βis the vector of coefficients of the explanatory
variables.
In our case, the event of interest is not ‘survival’ itself, but ‘being
dependency-free at a given age’. That is, we interpret function S(t,x)asthe
time spent free of dependency at a given age. At this point we must remind
that EDAD 2008 only contains records of alive people at 2008, hence the effect
of death is ignored. That is, the estimated being dependency-free probability
at a given age is in fact the probability of being dependency-free at a given
age given that a person is alive at that age. Thus, the next step consists in cor-
recting these estimates by survival probabilities given by the disabled Spanish
pensioners’ mortality table Orden TAS/4054/2005 (2005). Finally, we estimate
life expectancy at a given age xusing the following formula:
ˆ
ex=0.5 +
ωx
t=1
tˆ
px,
where tˆ
px=S(t,x)/S(0, x)andωis the limit of life.
We consider several scenarios to estimate LEFD attending to gender,
eight proximity-groups (IV–I for {dj<0}and I–IV for {dj0}) and three
dependency degrees (I-moderate, II-severe, III-major) given by the Spanish
legislation. Indeed, for a non-dependent person we compute three different
LEFDs, which are the expected number of years that a person can live outof
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70 I. ALBARRÁN ET AL.
each one of the three dependency degrees. Notice that the dependency history
of an individual may not reach all the states, that is, the first score reached
by an individual can be greater than 50 or 75 points. This is the reason why,
in the following tables and for the sake of simplicity, we call ‘degree I’ to the
expected number of years that a person can live out of any dependency degree
(score under 25 points); ‘degree II’ stands for the expected number of years that
a person can live out of severe or major dependency (score under 50 points);
‘degree III’ stands for the expected number of years that a person can live out
of major dependency (score under 75 points).
4.1. Searching for main effects on LEFD
The variables included in the Cox regression model are gender, proximity-
group and the disabilities recorded in EDAD 2008 related to dependency,
that is: to present difficulties in performing postural changes, bathing/hygiene,
control of physical needs, conducting household life, maintaining interaction
and interpersonal relationships, following medical treatments and mobility
difficulties (inside and outside the house).
We include in the analysis people that in 2008 were between 50 and 100
years old, with a dependency score of zero at the age of 30. We remind that
we are particularly interested in such a group of 30-year-old non-dependent
people in order to obtain LEFD estimates for different dependency scenarios,
which are useful for health, economic, demographic and insurance contexts.
We also remind that the sample is formed by 7446 individuals (2230 men and
5216 women), that represent more than one million Spanish people, according
to INE.
We estimate and validate Cox regression model with survival package
in R.
Figure 5contains a heat map of the main effects derived from Cox regres-
sion results estimated for each proximity-group and dependency degree (see
Table A2 in the Appendix for detailed results). Light colours represent low,
sometimes null, impacts on the LEFD, whereas dark colours stand for high
effects on the LEFD.
In the following we interpret the main variable effects on LEFD using
Figure 5. In both genders we observe that the loss of autonomy, that is,
the occurrence of disabilities, is more relevant for people with later onset of
dependency compared to a central trend. The variables with the most per-
sistent impact on both genders’ LEFD are control of physical needs and
bathing/hygiene. Additionally, medical treatments present a lasting effect on
men’s LEFD and the same happens on maintaining interaction and interper-
sonal relationships regarding women’s LEFD. On the other hand, mobility
difficulties inside and outside the house have the lowest impact on both gen-
ders’ LEFD. The second to low for men’s LEFD is maintaining interaction
and interpersonal relationships. Finally, from Table A2 in the Appendix, we
observe that the highest impact on men’s LEFD is registered by difficulties
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 71
FIGURE 5: Heat map of main effects derived from Cox regression results (based on Table A2 in Appendix).
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72 I. ALBARRÁN ET AL.
FIGURE 6: LEFD for men (panels a1–b4) and women (panels c1–d4). Age is represented in the horizontal
axis and years free of dependency in the vertical one.
in performing postural changes and by control of physical needs regard-
ing women’s LEFD. Both variables present a similar behaviour, having high
impacts on people with later onset of dependency compared to a central trend
and fading afterwards.
4.2. Contribution of the proximity-groups on estimated LEFD
In Figure 6we depict the estimated LEFD for the scenarios considered for the
eight proximity-groups for men and women. In particular, each panel contains
three curves corresponding to the evolution of the LEFD along age in three sit-
uations (At least Degree I, At least Degree II and Only Degree III).4In general,
looking at LEFD curves we can observe that, first, they reach higher values
for women than for men, meaning that the LEFD is higher for women than for
men; second, the decreasing rate is higher (and more abrupt) for men than for
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 73
women, which means that the loss of autonomy is higher (and more abrupt)
in men than in women; finally, we focus on the extreme proximity-groups
IV-below the deepest path (a4–c4 panels) and IV-above the deepest path (b4–
d4 panels). The former correspond to individuals with the best dependency
situation (latest onset of dependency), where we observe that LEFD curves
for ‘At least Degree I’ reach lower values for men than for women, indicating
that dependency phenomenon tends to appear earlier in men than in women.
Additionally, the estimation of women’s LEFD is very similar for the three
dependency situations (notice the proximity of the three curves), meaning that,
although the phenomenon tends to appear later in women than in men, its
intensity is higher in women than in men. A similar behaviour can be observed
in panels a3–c3 and a2–c2. The latter corresponds to individuals with the worst
dependency situation (earliest onset of dependency), where we can observe an
analogous shape of the LEFD for both genders in the three dependency situ-
ations, reaching slightly higher values for women, indicating a slightly worse
life expectancy for them. Additionally, we observe that the time spent free of
low or moderate dependency degree is quite similar in both genders. This may
suggest that for people with the earliest onset of dependency, having less than
50 points in the dependency score is crucial for spending (a maximum of) 8–10
years free of major dependency.
As a summary, Table 2contains the estimated LEFD for men and women
at three particular ages jointly with the LEFD calculated without taking into
account the partition by proximity-groups (rows LEFD for men and LEFD
for women). We may remind that these LEFD estimations are computed from
survey EDAD 2008 that contains only disabled people. Therefore, if we want
to extend these estimates to the Spanish population, they must interpreted
as a lower bound, that is, as the ‘at least’ expected numbers of years free of
dependency. Nevertheless, the methodology that we propose in this paper is
not restricted to this database.
In Table 2we observe that the variance of LEFD increases with age and
tends to decrease with dependency degree. In general, the variance is greater
for women. The relative errors5show evidence that the global LEFD by gen-
der, calculated without taking into account the partition by proximity-groups,
is far from any of the LEFD values estimated by proximity-groups. This means
that the global LFED may not be representative of the dependent Spanish
population, not even for those individuals within the most central proximity-
groups, that is, for those that are the nearest to the corresponding central
trend.
To illustrate the contribution of the partition by proximity-groups on the
LEFD estimation, we consider the following example. The global LEFD for
a 30-year-old women with dependency degree I is 21.6 years, which means
that the expected number of years that a 30-year-old woman can live out of
any dependency degree is 21.6. In other words, it is expected that a 30-year-
old woman can reach 51 years old out of any dependency degree. However,
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74 I. ALBARRÁN ET AL.
TABLE 2
SUMMARY OF LEFD FOR THREE PARTICULAR AGES.
Men 30yearsold 50yearsold 70yearsold
Degree I II III I II III I II III
{dj<0}Proximity-group IV 28.234 32.320 32.416 14.117 19.501 19.623 4.417 8.424 8.648
Proximity-group III 28.616 33.167 35.395 14.701 20.618 23.567 4.040 9.022 11.706
Proximity-group II 25.664 31.426 35.186 11.352 18.389 23.290 3.065 6.745 11.234
Proximity-group I 24.069 32.908 35.648 9.424 20.275 23.902 3.004 8.899 12.120
{dj0}Proximity-group I 21.848 26.600 35.375 7.378 12.228 23.541 1.838 3.711 11.812
Proximity-group II 20.792 25.213 31.548 6.524 10.375 18.474 1.346 2.857 7.400
Proximity-group III 14.762 22.884 31.427 4.695 8.647 18.659 0.878 2.127 7.425
Proximity-group IV 12.191 13.509 20.194 2.426 3.183 7.497 0.359 0.600 2.337
LEFD for men 18.733 23.614 31.868 6.475 10.429 19.391 1.556 3.195 8.583
Women 30 years old 50 years old 70 years old
Degree I II III I II III I II III
{dj<0}Proximity-group IV 32.805 34.370 35.099 20.195 22.210 23.175 8.187 9.780 11.110
Proximity-group III 31.714 33.353 35.264 18.694 20.864 23.394 6.465 8.783 11.589
Proximity-group II 30.578 33.549 35.614 17.215 21.141 23.860 4.753 8.973 12.116
Proximity-group I 26.172 33.743 11.557 21.380 2.343 8.699
{dj0}Proximity-group I 23.145 27.523 35.166 8.766 13.904 23.264 1.664 4.017 11.102
Proximity-group II 23.527 28.316 33.279 9.340 14.536 20.765 1.938 4.283 8.204
Proximity-group III 18.077 26.109 31.882 6.543 11.858 18.958 1.190 2.745 6.992
Proximity-group IV 13.852 14.923 21.651 3.288 3.909 9.114 0.579 0.759 2.757
LEFD for women 21.601 26.657 32.860 8.781 13.623 20.577 2.092 4.337 9.045
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 75
FIGURE 7: Gini index for men and women within the proximity-groups.
a more accurate estimation can be obtained by considering proximity-groups,
ranging from 32.8 years (proximity-group IV within set {dj<0}) to 13.8 years
(proximity-group IV within set {dj0}). That is, in the best situation, a
30-year-old woman can live out of any dependency degree until 62 years old
and, in the worst case, a 30-year-old woman becomes dependent at the age
of 43 years old. Notice that this difference of around 20 years is relevant,
at least, from demographic and economic points of view, in the sense that,
the expected dependent population would demand care services (health care,
pensions and other services) that should be covered and related expenditures
should be financed.
Finally, we can use Gini index as a numerical measure to summarize depen-
dency evolution per proximity-group. At this point it is important to remind
that dependency curves are associated to individuals that represent population
groups of different sizes.6Therefore, each individual is associated to a weight.
The sum of all of these weights is an estimation of the whole dependent Spanish
population.
Two variables are needed in order to calculate the Gini index: pvariable for
population and qvariable for quantities or amounts. In our case, pvariable has
been computed summing the weights of all individuals with positive score and
for each age ranging from 30 to 100 years old. On the other hand, qvariable has
been computed for each age ranging from 30 to 100 years old as the product of
(positive) scores by weights and represents the total amount of score values at
each age. We depict the results in Figure 7.
We can conclude the following. For proximity-groups within set {dj0},
the concentration tends to reduce as djincreases, which means that depen-
dency curves tend to be more similar as their distance to the deepest curve
increases, that is, the loss of autonomy in people with the earliest onset of
dependency tends to a singular extreme-pattern. The opposite happens for
proximity-groups within set {dj<0}, that is, Gini index tends to reduce as dj
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76 I. ALBARRÁN ET AL.
gets closer to zero, meaning that dependency curves tend to be less similar as
their distance to the deepest curve increases. This may suggest that the loss of
autonomy in people with the latest onset of dependency presents more than
one pattern (similar results were found in Albarrán-Lozano et al.,2017). Both
findings can be related with the heat map given in Figure 5. In particular, the
extreme-pattern observed for people with earliest onset of dependency corre-
sponds to few variable effects on LEFD (estimated via Cox regression model).
On the other hand, the diversity of patterns for people with later onset of
dependency can be associated to a wider range of variable effects on LEFD.
Finally, comparing both genders, women’s Gini index experiments a regular
decreasing, reaching always higher values than in men, meaning that women
have a wider range of dependency patterns than men.
5. CONCLUSIONS AND FURTHER RESEARCH
Dependency, that is, lack of autonomy in performing basic ADL, can be seen
as a consequence of the process of gradual aging. In Europe in general and
in Spain in particular this phenomenon represents a problem with economic,
political and social implications. The prevalence of dependency in the popula-
tion as well as its intensity and evolution over the course of a person’s life are
issues of greatest importance that should be addressed.
The aim of this work is to estimate LEFD, that is, the expected number
of years that a person can live free of this contingency based on mortality
and morbility conditions. The evolution of dependency in the disabled Spanish
population is studied through a pseudo panel constructed from EDAD 2008, in
the lack of longitudinal studies or the possibility to link different cross-sectional
surveys. In particular, individual dependency trajectories are obtained using
the retrospective reported information of each individual from birth up to
2008, contained in EDAD 2008, and applying the Spanish legislation (Act
39/2006 and Royal Decree 504/2007).
The main contribution of this paper is the estimation of LEFD based on
functional trajectories to enhance the regular estimation of health expectancy.
Using the information of EDAD 2008, we estimate the number of years
spent free of dependency for disabled people, according to gender, dependency
degree (moderate, severe, major) and the later or earlier onset of dependency
compared to a central trend.
In both genders we observe that the loss of autonomy, that is, the occur-
rence of disabilities, is more relevant for people with later onset of dependency
compared to the central trend. Control of physical needs and bathing/hygiene
present the most persistent impact on both genders’ LEFD, whereas the lowest
impact is registered by mobility difficulties inside and outside the house.
Concerning LEFD, we found that, first, in general, dependency-free life
expectancy is higher for women than for men. However, the intensity of
dependency is higher for those women with later onset of dependency. Second,
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 77
the loss of autonomy is higher (and more abrupt) in men than in women. Third,
for people with the earliest onset of dependency, having less than 50 points (out
of 100) in the dependency score is crucial for living a longer time free of major
dependency. Finally, the loss of autonomy in people with the earliest onset of
dependency tends to a singular extreme-pattern, characterized by few variable
effects on LEFD, whereas for people with later onset of dependency the loss of
autonomy has a diversity of patterns that can be associated to a wider range of
variable effects on LEFD. The diversity of theses patterns is higher in women
than in men.
The proposed methodology can be applied in several directions. For exam-
ple, dependency paths can be used in the health industry to help to discover
the true cost of different dependency patterns that can be established by
the proximity-groups and dependency degrees. Dependent population may
demand care services that should be offered and financed (public health and/or
private insurance companies).
ACKNOWLEDGEMENTS
Financial support is from research project MTM2014-56535-R by the Spanish
Ministry of Economy and Competitiveness. The authors thank the Associate
Editor and two Referees, whose comments helped to improve the quality of the
paper.
NOTES
1.Resolution R(98) of the Council of Europe defines dependency as ‘such state in which peo-
ple, whom for reason connected to the lack or loss of physical, mental or intellectual autonomy,
require assistance and/or extensive help in order to carry out common everyday actions’. This
definition has been translated into national legislations in a heterogeneous way (Kamette, 2011).
2.Three surveys about disability have been undertaken by the Spanish Statistical Office
(INE) during the last 30 years. The first one was conducted in 1986 and was the Survey about
Disabilities, Impairments and Handicaps. Then came the Survey about Disabilities, Impairments
and Health Status, which was prepared using data from 1999. Finally, the last one was EDAD in
2008. Although all of them talk about disabilities, it is not possible to track this phenomenon in
a homogeneous way along the years because the definition of that concept changed through the
years depending on the classification that was used to prepare the survey.
3.In 2001, the World Health Organization (2011a) established a framework for measur-
ing health and disability at both individual and population levels, which was known as the
‘International classification of functioning (ICF), disability and health’. The ICF tries to establish
a consensus in its understanding, by establishing a difference between the basic activities of living
daily (ADL) and the instrumental ADL. The basic activities are defined as those activities which
are essential for an independent life.
4.‘At least Degree I’ stands for expected number of years that a person can live out of any
dependency degree (score under 25 points); ‘At least Degree II’ stands for the expected num-
ber of years that a person can live out of severe or major dependency (score under 50 points);
‘Only Degree III’ stands for the expected number of years that a person can live out of major
dependency (score under 75 points).
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78 I. ALBARRÁN ET AL.
5.The relative error of a given LEFD is computed as its difference to the global LEFD by
gender divided by the global LEFD by gender.
6.In EDAD 2008 a two-stage sampling was conducted by INE, leading to individuals that
represent population groups of different sizes.
REFERENCES
ALBARRÁN,I.andALONSO, P. (2009) La población dependiente en España: estimación del
número y coste global asociado a su cuidado. Estudios de Economía,36(2), 127–163.
ALBARRÁN,I.,ALONSO,P.,ARRIBAS-GIL,A.andGRANÉ, A. (2014) Can personal dependency
paths help to estimate life expectancy free of dependency? In Mathematical and Statistical
Methods for Actuarial Sciences and Finance (eds. M. Sibilio and C. Perna), pp. 1–5. Cham,
Switzerland: Springer.
ALBARRÁN,I.,ALONSO,P.andGRANÉ, A. (2015) Profile identification via weighted related metric
scaling: An application to dependent Spanish children. Journal of the Royal Statistical Society
A,178, 1–26.
ALBARRÁN-LOZANO,I.,ALONSO-GONZÁLEZ,P.andARRIBAS-GIL, A. (2017) Dependence evolu-
tion in the Spanish disabled population: A functional data analysis approach. Journal of the
Royal Statistical Society A,180(2), 657–677.
ARRIBAS-GIL,A.andROMO, J. (2012) Robust depth-based estimation in the time warping model.
Biostatistics,13, 398–414.
BIESSY, G. (2017). Continuous-time semi-Markov inference of biometric laws associated with a
long-term care insurance portfolio. Astin Bulletin,47, 527–561.
COX, D. (1972) Regression models and life tables. Journal of the Royal Statistical Society B,34,
187–220.
COX,D.R.andOAKES, D. (1984) Analysis of Survival Data. London: Chapman & Hall.
CZADO,C.andRUDOLPH, F. (2002) Application of survival analysis methods to long-term care
insurance. Insurance: Mathematics and Economics,31, 395–413.
DUPUY,J.,LOUBES,J.andMAZA, E. (2011) Non parametric estimation of the structural
expectation of a stochastic increasing function. Statistics and Computing,21(1), 121–136.
EUROSTAT (2009) Health statistics and atlas on mortality in the European Union. Eurostat.
Luxembourg.
FONG,J.,SHERRIS,M.andYAP, J. (2017) Forecasting disability: Application of a frailty model.
Scandinavian Actuarial Journal,2017(2), 125–147.
FRIES, J. (1983) The compression of morbidity. The Milbank Memorial Fund Quarterly,61, 397–
419.
GASSER,T.andKNEIP, A. (1995) Searching for structure in curve samples. Journal of the American
Statistical Association,90, 1179–1188.
GASSER,T.,MÜLLER,H.,KÖHLER,W.,MOLINARI,L.andPRADER, A. (1984) Nonparametric
regression analysis of growth curves. The Annals of Statistics,12, 210–229.
INE (2010) Encuesta sobre Discapacidad, Autonomía personal y Situaciones de Dependencia
(EDAD), Metodología. Ed. Subdirección General de Estadísticas Sociales Sectoriales (INE),
Madrid, España.
KAMETTE, F. (2011) Dependency care in the EU: A comparative analysis. Foundation Robert
Schumann. Social Issues European Issue, No. 196.
KNEIP,A.andGASSER, T. (1992) Statistical tools to analyze data representing a sample of curves.
The Annals of Statistics,16, 82–112.
LEVANTESI,S.andMENZIETTI, M. (2018) Natural hedging in long-term care insurance. Astin
Bulletin,48(1), 233–274.
LIU,X.andMÜLLER, H. (2004) Functional convex averaging and synchronization for time-
warped random curves. Journal of the American Statistical Association,99(467), 687–699.
LLOYD-SHERLOCK,P.,MCKEE,M.,EBRAHIM,S.,GORMAN,M.,GREEGROSS,S.,PRINCE,M.,
PRUCHNO,R.,GUTMAN,G.,KIRWOOD,T.,ONEILL,D.,FERRUCCI,L.,KRITCHEWSKI,S.and
VELLAS, B. (2012) Population ageing and health. The Lancet,379, 1295–1296.
, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/asb.2018.34
Downloaded from https://www.cambridge.org/core. IP address: 88.15.167.187, on 08 Mar 2019 at 10:15:29, subject to the Cambridge Core terms of use
HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 79
LÓPEZ-PINTADO,S.andROMO, J. (2009) On the concept of depth for functional data. Journal of
the American Statistical Association,114, 486–503.
MARTEL,L.andBÉLANGER, A. (2000) Regression models and life tables. Canadian Social Trends-
Statistics Canada Autumn,2000, 26–29.
ORDEN TAS/4054/2005 (2005) BOE num. 316, 28 de diciembre de 2005. Ministerio de Trabajo y
Asuntos Sociales.
RAMSAY,J.andSILVERMAN, B. (2005) Functional Data Analysis, 2nd ed. New York: Springer
Series in Statistics.
ROBINE,J.andRITCHE, K. (1991) Healthy life expectancy: Evaluation of global indicator of
change in population health. BMJ,302, 457–460.
ROBINE,J.,JAGGER,C.,MATHERS,C.,KRIMMINS,E.andSUZMAN, R. (2003) Determining Health
Expectancies. Chichester: Wiley.
SANDERSON,W.andSCHERBOV, S. (2010) Remeasuring aging. Science,329, 1287–1288.
TARABELLONI,N.,ARRIBAS-GIL,A.,IEVA,F.,PAGANONI,A.andROMO, J. (2016) roahd: Robust
analysis of high dimensional data. Package on CRAN. Version 1.0.4. Published 2016-07-06.
WANG,K.andGASSER, T. (1997) Alignment of curves by dynamic time warping. The Annals of
Statistics,25, 1251–1276.
WHO (2011a) Global health and aging. WHO (World Health Organization). US National
Institute of Aging.
WHO (2011b) Global report on disability. WHO (World Health Organization). US National
Institute of Aging.
IRENE ALBARRÁN
Statistics Department
Universidad Carlos III de Madrid
C/Madrid 126, 28903 Getafe
Spain
E-Mail: irene.albarran@uc3m.es
PABLO J. ALONSO-GONZÁLEZ
Economics Department
Universidad de Alcalá
Plaza de la Victoria 2, 28802 Alcalá de Henares
Spain
E-Mail: pablo.alonsog@uah.es
ANA ARRIBAS-GIL
UC3M-BS Institute of Financial Big Data
C/Madrid 135, 28903 Getafe
Spain
E-Mail: ana.arribas@uc3m.es
AUREA GRANÉ (Corresponding author)
Statistics Department
Universidad Carlos III de Madrid
C/Madrid 126, 28903 Getafe
Spain
E-Mail: aurea.grane@uc3m.es
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80 I. ALBARRÁN ET AL.
APPENDIX
TABLE A1
DEPENDENCY GRADUATION ACCORDING TO SPANISH LEGISLATION.
Dependency Degree Level Score Dependency Degree Level Score
Non-dependent [0, 25) Severe II 1 [50, 65)
II 2 [65, 75)
Moderate I 1 [25, 40) Major III 1 [75, 90)
I 2 [40, 50) III 2 [90, 100]
Moderate dependency The person needs help in order to perform various basic ADLa
at least once a day or the person needs intermittent or limited
support for his/her personal autonomy.
Severe dependency The person needs help in order to perform various basic ADL
two or three times a day, but he/she does not want the
permanent support of a carer or he/she needs extensive support
for his/her personal autonomy.
Major dependency The person needs help in order to perform various basic ADL
several times a day or he/she needs the indispensable and
continuous support of another person or he/she needs
generalized support for his/her personal autonomy.
aADL stands for Activities of Daily Living.
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 81
TABLE A2
COX REGRESSION RESULTS FOR MEN AND WOMEN:eβSANDβS STANDARD DEVIATION (WITHIN PARENTHESIS).
Men Set {dj<0}; Individual trajectories below the deepest path
Proximity-group IV Proximity-group III Proximity-group II Proximity-group I
Degree I II III I II III I II III I II III(*)
postural 1.283 1.566 3.107 1.496 2.321 17.951 1.120 3.094 5.323 2.240 4.678
changes (0.02) (0.026) (0.047) (0.014) (0.02) (0.068) (0.016) (0.031) (0.102) (0.015) (0.029)
mobility 1.142 0.862 1.839 1.202 0.225 0.219 1.406 0.416 5.250 1.622 0.310
(0.028) (0.029) (0.045) (0.026) (0.027) (0.064) (0.017) (0.032) (0.139) (0.017) (0.036)
bathing/ 2.966 4.844 0.000 1.090 7.189 0.000 2.117 3.040 0.000 1.334 2.046
hygiene (0.044) (0.106) (0) (0.024) (0.067) (0) (0.018) (0.04) (0.06) (0.016) (0.037)
physical 0.770 1.548 5.162 0.750 1.287 4.222 0.648 1.105 0.989 1.340 2.845
needs (0.019) (0.027) (0.046) (0.017) (0.023) (0.066) (0.018) (0.026) (0.06) (0.017) (0.03)
medical 1.471 14.768 0.000 1.359 4.597 0.000 1.225 2.830 0.000 1.419 2.407
treatments (0.025) (0.086) (0) (0.018) (0.043) (0) (0.016) (0.034) (0) (0.016) (0.037)
household 2.116 0.603 0.000 3.164 3.407 6.896 1.835 1.756 0.000 1.292 1.817
life (0.04) (0.045) (0) (0.032) (0.064) (0.106) (0.019) (0.033) (0) (0.017) (0.037)
interpers. 1.189 1.521 0.000 1.555 2.667 3.091 0.797 1.178 0.000 2.134 1.317
rel. (0.017) (0.02) (0) (0.021) (0.023) (0.042) (0.019) (0.028) (0) (0.021) (0.035)
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82 I. ALBARRÁN ET AL.
TABLE A2
CONTINUED.
Men Set {dj0}; Individual trajectories above the deepest path
Proximity-group I Proximity-group II Proximity-group III Proximity-group IV
Degree IIIIIIIIIIIIIIIIIIIIIIII
postural 1.183 1.547 3.163 1.304 1.580 2.689 1.161 1.320 1.779 0.592 0.816 1.787
changes (0.01) (0.012) (0.048) (0.011) (0.014) (0.025) (0.01) (0.012) (0.022) (0.012) (0.013) (0.019)
mobility 0.674 0.491 0.178 1.057 0.985 0.529 0.789 0.781 1.045 0.711 1.479 0.626
(0.011) (0.013) (0.046) (0.012) (0.016) (0.023) (0.012) (0.013) (0.021) (0.014) (0.016) (0.02)
bathing/ 1.408 2.657 0.000 1.256 0.829 0.319 1.337 1.361 14.387 1.020 1.002 2.836
hygiene (0.016) (0.024) (0) (0.015) (0.02) (0.035) (0.015) (0.018) (0.119) (0.019) (0.021) (0.038)
physical 0.523 0.904 2.190 0.544 1.807 5.502 0.386 1.136 5.619 0.437 0.930 4.103
needs (0.01) (0.012) (0.041) (0.011) (0.014) (0.037) (0.011) (0.012) (0.034) (0.011) (0.012) (0.022)
medical 1.488 1.994 363.577 1.094 1.156 1.780 1.309 2.453 0.520 1.116 3.600 1.022
treatments (0.013) (0.017) (0.395) (0.013) (0.017) (0.038) (0.014) (0.017) (0.028) (0.016) (0.021) (0.022)
household 1.325 1.073 0.539 1.226 1.342 4.731 1.054 1.214 1.759 1.041 0.755 1.130
life (0.013) (0.016) (0.047) (0.013) (0.017) (0.053) (0.013) (0.016) (0.028) (0.017) (0.019) (0.025)
interpers. 1.264 2.045 1.901 0.935 1.205 1.872 0.680 0.793 1.818 0.753 0.939 1.995
rel. (0.011) (0.014) (0.037) (0.011) (0.012) (0.017) (0.01) (0.011) (0.015) (0.01) (0.01) (0.013)
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HOW FUNCTIONAL DATA CAN ENHANCE THE ESTIMATION OF HEALTH EXPECTANCY 83
TABLE A2
CONTINUED.
Women Set {dj<0}; Individual trajectories below the deepest path
Proximity-group IV Proximity-group III Proximity-group II Proximity-group I
Degree I II III I II III I II III I II III(*)
postural 1.069 1.536 3.633 1.709 2.841 2.599 1.027 3.249 11.905 1.032 1.704
changes (0.012) (0.019) (0.028) (0.01) (0.016) (0.029) (0.01) (0.018) (0.112) (0.008) (0.017)
mobility 1.104 0.472 0.349 1.242 0.713 0.648 1.332 0.723 0.000 1.054 0.444
(0.019) (0.024) (0.026) (0.012) (0.016) (0.027) (0.011) (0.016) (0) (0.008) (0.017)
bathing/ 1.344 2.968 3.718 1.312 1.094 4.165 1.523 2.320 0.000 0.831 1.250
hygiene (0.02) (0.039) (0.063) (0.014) (0.026) (0.107) (0.011) (0.027) (0) (0.01) (0.024)
physical 1.292 5.036 11.112 1.508 2.805 5.735 0.701 1.444 23.578 0.856 1.631
needs (0.013) (0.02) (0.038) (0.01) (0.014) (0.035) (0.011) (0.015) (0.093) (0.01) (0.017)
medical 1.399 1.255 0.000 0.880 3.403 0.000 0.901 2.711 1.613 0.963 2.802
treatments (0.016) (0.024) (0) (0.011) (0.023) (0) (0.01) (0.02) (0.055) (0.008) (0.021)
household 3.097 2.190 0.000 2.640 1.433 1.793 3.350 1.627 0.000 1.650 0.926
life (0.025) (0.036) (0) (0.018) (0.028) (0.079) (0.018) (0.029) (0) (0.012) (0.023)
interpers. 1.133 1.438 3.128 0.819 1.142 3.460 0.795 1.038 1.122 0.769 1.082
rel. (0.01) (0.012) (0.015) (0.011) (0.013) (0.024) (0.012) (0.015) (0.035) (0.013) (0.021)
(continued)
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84 I. ALBARRÁN ET AL.
TABLE A2
CONTINUED.
Women Set {dj0}; Individual trajectories above the deepest path
Proximity-group I Proximity-group II Proximity-group III Proximity-group IV
Degree IIIIIIIIIIIIIIIIIIIIIIII
postural 1.001 1.734 2.640 1.241 1.510 1.366 1.551 1.459 2.161 1.271 1.062 1.902
changes (0.006) (0.009) (0.026) (0.007) (0.009) (0.016) (0.007) (0.009) (0.015) (0.008) (0.008) (0.013)
mobility 0.787 0.644 0.716 0.918 0.933 1.059 0.639 0.902 0.618 0.987 0.930 1.373
(0.007) (0.008) (0.022) (0.007) (0.009) (0.015) (0.008) (0.009) (0.013) (0.009) (0.009) (0.013)
bathing/ 1.025 1.461 1.076 1.196 1.373 2.488 0.833 0.698 3.224 0.596 1.324 0.706
hygiene (0.008) (0.012) (0.044) (0.009) (0.012) (0.03) (0.01) (0.011) (0.027) (0.014) (0.015) (0.026)
physical 0.653 1.012 5.079 0.620 1.712 2.395 0.490 1.356 2.003 0.573 1.108 2.085
needs (0.006) (0.008) (0.029) (0.007) (0.009) (0.018) (0.007) (0.008) (0.015) (0.007) (0.008) (0.011)
medical 0.973 1.857 0.468 0.910 1.916 1.046 0.859 1.612 0.584 0.969 1.988 0.884
treatments (0.007) (0.009) (0.027) (0.008) (0.012) (0.017) (0.008) (0.01) (0.015) (0.01) (0.01) (0.013)
household 1.395 0.722 0.935 1.791 0.708 1.566 1.489 1.304 1.538 1.231 0.928 1.823
life (0.01) (0.011) (0.039) (0.01) (0.012) (0.031) (0.01) (0.011) (0.022) (0.012) (0.011) (0.021)
interpers. 0.769 1.308 5.747 0.745 1.048 2.212 0.696 0.890 2.695 0.701 0.760 2.250
rel. (0.008) (0.009) (0.022) (0.007) (0.008) (0.011) (0.007) (0.007) (0.01) (0.006) (0.006) (0.008)
()The sample size is not enough to obtain significative results.
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