@let@token Cause-of-Death Mortality: What Can Be Learned ...
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Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? Héloïse Labit Hardy 1 PhD Student, University of Lausanne, Switzerland joint work with S. Arnold (-Gaille) 1 , A. Boumezoued 2 , N. El Karoui 2 available at https://hal.archives-ouvertes.fr/hal-01157900 IAA Conference, Oslo, Norway June 10, 2015 1 Department of Actuarial Science, Faculty of Business and Economics - Extranef, University of Lausanne, CH-1015 Lausanne, Switzerland [email protected], [email protected] 2 Probability and Random Models Laboratory, Pierre and Marie Curie University, 4 place Jussieu, 75005 Paris, France [email protected], [email protected] Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 1/26
Transcript of @let@token Cause-of-Death Mortality: What Can Be Learned ...
Cause-of-Death Mortality: What Can BeLearned From Population Dynamics?
Héloïse Labit Hardy 1
PhD Student, University of Lausanne, Switzerland
joint work with S. Arnold (-Gaille)1, A. Boumezoued2, N. El Karoui2
available at https://hal.archives-ouvertes.fr/hal-01157900
IAA Conference, Oslo, NorwayJune 10, 2015
1Department of Actuarial Science, Faculty ofBusiness and Economics - Extranef, University ofLausanne, CH-1015 Lausanne, Switzerland
[email protected], [email protected]
2Probability and Random Models Laboratory, Pierreand Marie Curie University, 4 place Jussieu, 75005Paris, France
[email protected],[email protected]
Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 1/26
Introduction
0 20 40 60 80
−10
−8
−6
−4
−2
0
French death rates for males between 1960 and 2000 for external causes
Age
ln m
u
19601970198019902000
0 20 40 60 80
−10
−8
−6
−4
−2
0
French death rates for males between 1960 and 2000 for cancers
Age
ln m
u
19601970198019902000
Source : The World Health Organization (WHO)
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Introduction
Objective :
Study impacts of changes in cause-of-death mortality on the wholepopulation age structure
ä Model population dynamics
. By taking into account deaths and births :
� with birth and death rates depending on genderand age, invariant over time
. Reference : Bensusan, Boumezoued, El Karoui and Loisel(working paper)
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
1.1 Competing risks model1.2 Dependence assumptions
1. Cause-of-Death Mortality1.1 Competing risks model1.2 Dependence assumptions
2. Population Dynamics Model
3. Application to French data
Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 4/26
1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
1.1 Competing risks model1.2 Dependence assumptions
Let us consider two causes of death, A and B , modelled by twocompetiting clocks :
ä τA : the length of time for clock A
ä τB : the length of time for clock B
Lifetime τ of an individual :
τ = min(τA, τB)
Remark :
ä if A only possible cause, τA : lifetime of the individual
ä if B only possible cause, τB : lifetime of the individual
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
1.1 Competing risks model1.2 Dependence assumptions
ä In practice, only the lifetime τ = min(τA, τB) is observed :
The crude force of mortality for cause A :
µ̃A(a) =P(a < τ ≤ a + da, τ = τA | τ > a)
da
when causes A and B are in competition
ä We are interested by :
The net force of mortality for cause A :
µA(a) =P(a < τA ≤ a + da | τA > a)
da
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
1.1 Competing risks model1.2 Dependence assumptions
Dependence structure between causes of deathForces of mortality :
µA(a) 6= µ̃A(a)
µB(a) 6= µ̃B(a)
ä Dependence structure non-identifiable from available data
⇒ Assumptions on dependence between causes of death
ä Exemples of dependence structure :
. Independence assumption : Chiang et al. (1968)
. Independence assumption modified : Alai et al. (2013)
. Dependence modelled by copulas : Carriere (1994),Kaishev et al. (2007), Dimitrova et al. (2013)
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
1.1 Competing risks model1.2 Dependence assumptions
Independence assumption between causes of death
ä Assumption widely used
. ex : Prentice et al. (1978), Tsai et al. (1978), Willmoth(1995), Putter et al. (2007)
ä Highlight interesting effects
ä Simplification of the modelling :
. Under independence :
� µA(a) = µ̃A(a), µB(a) = µ̃B(a)
� Remove cause B : µ̃A(a)
� Reduce cause B : µ′B(a) = α.µ̃B(a), α ∈ (0, 1)
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
1. Cause-of-Death Mortality
2. Population Dynamics Model2.1 Deterministic equation2.2 Stochastic simulations
3. Application to French data
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
ä Deterministic approach :
. First population dynamics models
� McKendrick (1926), Von Foerster (1959)
. Model evolution of the average population age pyramid
� Age dependency ratio of the average population : rt
ä Stochastic approach :
. Provides scenarios of the population age pyramid
� Average age dependency ratio : E[Rt ] (non-linearquantity)
ä In large population : rt approximation of Rt
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
ä The population structure described by the vector :
g(a, t) =(
g(f , a, t)g(m, a, t)
)
. g(a, t) : average number of individual with age a at time t
ä The population dynamics is defined by :
. Deaths : ( ∂a + ∂t)g(a, t) = −(µf (a) 00 µm(a)
)g(a, t)
. Births : g(0, t) =(∫
R+g(f , a, t)bf (a) da
)( p1− p
)p : probability for a newborn to be a female
Remark : population without migrationHéloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 11/26
1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
•N0
T0 = 0
• Start at T0 with population N0population size
time
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
T0 = 0
•N0
• Start at T0 with population N0population size
timeT1
• T1 = T0 + γ, with γ exp. r.v.
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
T0 = 0
•N0
• Start at T0 with population N0population size
timeT1
• T1 = T0 + γ, with γ exp. r.v.• Select an individual (εi , ai ) uniformly
and identify the event
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
T0 = 0
•N0
• Start at T0 with population N0population size
timeT1
• T1 = T0 + γ, with γ exp. r.v.• Select an individual (εi , ai ) uniformly
and identify the event :- Birth : add a new individual N1 = N0 + 1•
•
N1
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
T0 = 0
•N0
• Start at T0 with population N0population size
timeT1
• T1 = T0 + γ, with γ exp. r.v.• Select an individual (εi , ai ) uniformlyand identify the event :- Birth : add a new individual N1 = N0 + 1- Death : remove the individual N1 = N0 − 1
•
•
N1
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
T0 = 0
•N0
• Start at T0 with population N0population size
timeT1
• T1 = T0 + γ, with γ exp. r.v.• Select an individual (εi , ai ) uniformlyand identify the event :- Birth : add a new individual N1 = N0 + 1- Death : remove the individual N1 = N0 − 1- No event : N1 = N0•N1
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
2.1 Deterministic equation2.2 Stochastic simulations
T0 = 0
•N0
population size
timeT1
No event
•N1
T2
Birth
•N2
T3
Death
•N3
T4
Death
•N4
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
3.1 Scenarios : All causes3.2 Scenarios : Cause removal3.3 Scenarios : Cause reduction
1. Cause-of-Death Mortality
2. Population Dynamics Model
3. Application to French data3.1 Scenarios : All causes3.2 Scenarios : Cause removal3.3 Scenarios : Cause reduction
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1. Cause-of-Death Mortality2. Population Dynamics Model3. Application to French data
3.1 Scenarios : All causes3.2 Scenarios : Cause removal3.3 Scenarios : Cause reduction
Population dynamics : 10 000 individuals, 100 years horizon
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Population dynamics : 100 000 individuals, 100 years horizon
Age pyramidAge pyramid
1000 500 0 500 1000
07
1524
3342
5160
6978
8796
107
119
Number of males Number of females
Initial population in 2008Final men population in 2108Final women population in 2108
0 20 40 60 80 100
2530
3540
4550
Age Dependency ratio from 2008 to 2108 (10 scenarios)
Time
Dep
ende
ncy
ratio
(%
)
All causes
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Cause removal : Cancers
0 20 40 60 80 100
2530
3540
4550
Age Dependency ratio from 2008 to 2108 (10 scenarios)
Time
Dep
ende
ncy
ratio
(%
)
Ignoring cancersAll causes
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Cause removal : External causes
0 20 40 60 80 100
2530
3540
4550
Age Dependency ratio from 2008 to 2108 (10 scenarios)
Time
Dep
ende
ncy
ratio
(%
)
Ignoring external causesAll causes
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Cause reduction : Deterministic age dependency ratio
0 20 40 60 80 100
2530
3540
4550
Time (years)
Dep
ende
nce
ratio
(%
)
Ignoring cancers (e0h=82.2, e0f=87.7)Reduction of cancers (e0h=79.1, e0f=85.1)Ignoring external causes (e0h=79.1, e0f=85.1)All causes (e0h=77.7, e0f=84.4)
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Conclusion
ä With a population dynamics model, we study impacts ofcause-of-death reductions on the population age structure :
. Model deaths and births, with rates invariant over time
. Under independence assumption between causes of death
⇒ Study the whole population dynamics give additionalinformations :
. With the same life expectancy at birth, cause reductionscan have different impacts on the age dependency ratio
ä Following :. Study dependence between causes of death. Take into account mortality improvement over the time. Study cause-of-death mortality by age, gender and
socio-economic variable
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Bibliographie
[1] Alai, D.H., Arnold(-Gaille), S., Sherris, M. (2015) Modelling Cause-of-DeathMortality and the Impact of Cause-Elimination. Annals of Actuarial Science
[2] Arnold, S., Boumezoued, A., Labit Hardy, H., El Karoui., N. (2015)Cause-of-Death Mortality : What Can Be Learned From Population Dynamics ?Working paper, https ://hal.archives-ouvertes.fr/hal-01157900
[3] Bensusan, H. (2010) Risques de taux et de longévité : Modélisationdynamique et applications aux produits dérivés et á l’assurance vie. PhDthesis, Ecole Polytechnique
[4] Chiang, C. L. (1968) Introduction to Stochastic Process in Biostatistics.John Wiley and Sons, New York
[5] Bensusan, H., A. Boumezoued, N. El Karoui, S. Loisel. Impact ofheterogeneity in human population dynamics. working paper
[6] Tran, V.C. (2006) Modéles particulaires stochastiques pour des problémesd’évolution adaptative et pour l’approximation de solutions statistiques. PhDthesis, Université Paris X - Nanterre
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