Interpreting mortality trends in the presence of heterogeneity ......Modeling heterogeneous...

Interpreting mortality trends in the presence of

heterogeneity: A population dynamics approach

Sarah Kaakaı

Laboratoire Manceau de Mathematiques, Le Mans Universite

Based on joint works with S. Arnold, N. El Karoui, H. Labit-Hardy and

A. Villegas

Longevity Heterogeneity and Pension Design workshop, 28 january 2020

Outline

1 Introduction

2 How can a cause-of-death reduction be compensated for by the population

heterogeneity? A dynamic approach.

3 Consistence of mortality forecast

1

Socioeconomic gradient in mortality

§ Research on the relationship between socioeconomic status (SES) and

mortality is longstanding (Villerme (1830), General Register Office (1851))

ñ broad consensus on the strong correlation between SES and

mortality.

§ New trends observed in the past decades: increasing of socioeconomic

gaps in health and mortality.

US female life expectancy gaps at birth between less and more educated

women: 7.7 years in 1990, 10.3 years in 2008 (Olshansky et al. (2012)).

Gap in male life expectancy at 65 between higher managerial and routine

occupations (England Wales): 2.4 years 1982-1986, 3.9 years 2007-2011,

ONS).

Widening gaps: socioeconomic subgroups experience rather different

mortality than national mortality rates.

2

Taking heterogeneity into account

§ SES inequities declared key public issues by the World Health

Organization in its last report on ageing and health.

§ Not taking into account heterogeneity can lead to:

Increased inequalities due public health reforms (Alai et al. (2017)) or

“unfair” redistribution properties of pensions systems.

Errors in funding of annuity and pension obligation (Meyricke and

Sherris (2013), Villegas and Haberman (2014)).

§ Better understanding of heterogeneity allows for a better understanding

of basis risk (Longevity basis risk report (2014))

3

Modeling heterogeneous mortality rates

§ Growing literature in the joint modeling and forecasting of the

mortality of socioeconomic subgroups Bensusan (2010), Jarner and

Kryger (2011), Villegas and Haberman (2014), Cairns et al. (2016) ...

§ Remaining questions:

Interpreting targets set by institutions (Department of Health, WHO)

(Alai et al. (2017)).

Consistency of sub-national and national estimates/forecasts (Shang

and Hyndman (2017), Shang and Haberman (2017)).

§ Standard tool for modelling and forecasting longevity: mortality rates.

§ Approach: take into account all population data rather that just

mortality data.

4

What can we learn from population dynamic?

Population divided in p risk classes:

§ One year central death rate in the global population for individuals age

x during calendar year t:

µxt “Dxt

Ext“

pÿ

j“1

D jxt

Ext

The relative exposure W jxt “

E jxt

Extis linked to the proportion of

individual of age x in risk class j .

Evolution of these quantities are determined by the population

dynamics and can vary a lot depending on the age x and time t.

How changes in the socioeconomic composition of the population affect

aggregated indicators? Could we miss a cause-of-death reduction in

presence of heterogeneity?

5





µxt “Dxt

Ext“

pÿ

j“1

D jxt

Ext“

pÿ

j“1

D jxtE

jxt

E jxtExt


E jxt








5





µxt “Dxt

Ext“

pÿ

j“1

D jxt

Ext“

pÿ

j“1

µjxt

E jxt

Ext


E jxt








5

Outline

1 Introduction




6

Data

§ Two datasets:

1981-2007: Department of Applied Health Research, UCL.

2001-2015: Office for National Statistics, UK .

§ English cause-specific number of deaths and mid-year population

estimates per socioeconomic circumstances, age and gender.

Socioeconomic circumstances are measured by the Index of multiple

deprivation (IMD), based on the postcode of individuals.

§ Small areas (LSOA) are ranked based on seven broad criteria: income,

employment, health, education, barriers to housing and services, living

environment and crime.

§ This ranking permits to divide the population in 5 quintiles with about

same number of individuals in each quintile.

7

Life expectancy

Figure: Evolution of period life expectancy at age 65

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Females

1980 1985 1990 1995 2000 2005 2010 2015

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14

16

18

20

22

Year

Per

iode

life

exp

ecta

ncy

IMD Quintiles●

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1

2

3

4

5

6

(a) Females

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Males

1980 1985 1990 1995 2000 2005 2010 2015

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14

16

18

20

22

Year

Per

iode

life

exp

ecta

ncy

IMD Quintiles●

●

●

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●

●

1

2

3

4

5

6

(b) Males

§ Increase in gap: 2.2 ñ 4.2 years (females), 2.9 ñ 3.9 (males).

8

Mortality by deprivation

Figure: Males average annual mortality improvement rates by age

−2.5%

−2%

−1.5%

−1%

−0.5%

0%

0.5%

1%

1.5%

2%

2.5%

3%

3.5%

25 30 35 40 45 50 55 60 65 70 75 80

Age

Ave

rage

d an

nual

impr

ovem

ent r

ates

IMD Quintiles1

2

3

4

5

6

(a) 1981-1995

−2.5%

−2%

−1.5%

−1%

−0.5%

0%

0.5%

1%

1.5%

2%

2.5%

3%

3.5%

25 30 35 40 45 50 55 60 65 70 75 80

Age

Ave

rage

d an

nual

impr

ovem

ent r

ates

IMD Quintiles1

2

3

4

5

6

(b) 2001-2015

§ 2001-2015: widening of mortality improvement rates gap above 60.

9

Population composition, 2015

Age-pyramids by IMD quintile, 2015

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100+

20152010

20052000

19951990

19851980

1975

19701965

19601955

19501945

19401935

19301925

19201915

100 80 60 40 20 0 20 40 60 80 100Number of individuals (in thousands)

Age

Year of birth

Type Males Females

(a) Most deprived quintile ($)

Median age: 33y

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100+

20152010

20052000

19951990

19851980

1975

19701965

19601955

19501945

19401935

19301925

19201915

100 80 60 40 20 0 20 40 60 80 100Number of individuals (in thousands)

Age

Year of birth

Type Males Females

(b) Least deprived quintile ($$$$$)

Median age: 44.2y

§ Baby-boom cohort less deprived than younger/older cohorts.10

Changes in the population composition

Figure: Compostion of males age classes in years 1981, 1990, 2005, 2015.

18%

20%21%20%

21%

19%

21%21%20%20%

22%22%21%

18%

17%

23%23%

21%

17%

14%

Age 65−74 in 1981 (Cohorts 1907−1916)

Age 65−74 in 1990 (Cohorts 1916−1925)

Age 65−74 in 2005 (Cohorts 1931−1940)

Age 65−74 in 2015 (Cohorts 1941−1950)

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

0

10

20

Pro

port

ion

of p

opul

atio

n

IMD Quintiles 1 2 3 4 5

(a) Age class 65-74

17%

19%

20%22%22%

17%

19%

20%

22%22%

16%

18%

20%

23%22%

14%

17%

20%

25%24%

Age 25−34 in 1981 (Cohorts 1947−1956)

Age 25−34 in 1990 (Cohorts 1956−1965)

Age 25−34 in 2005 (Cohorts 1971−1980)

Age 25−34 in 2015 (Cohorts 1981−1990)

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

0

10

20

Pro

port

ion

of p

opul

atio

n

IMD Quintiles 1 2 3 4 5

(b) Age class 25-34

§ Decrease of deprivation over time for older age classes.

§ Increase of deprivation for younger age classes.

11

Heterogeneous population dynamics

§ Simple age-structured population dynamics framework to illustrate

different impacts of heterogeneity on the aggregated mortality.

§ Deterministic evolution of each subgroup is described by a

McKendrick (1926) -Von Foerster (1959).

§ Equation for each gender ε “ m or f and subgroup:

Aging law:

pBa ` Btqgεj pa, tq “ ´µ

εj pa, tqg

εj pa, tq

Birth law:

g εj p0, tq “

ż a:

0

pεg fj pa, tqbj pa, tqda

Initial Pyramid:

g εj pa, 0q

12

Aggregated population

§ Aggregated population:

g εpa, tq “

řpj“1 g

εj pa, tq

Aging law: pBa ` Btqgεpa, tq “ ´dε

pa, tqg εpa, tq

§ Aggregated death rate:

Weighted sum of the subpopulations death rates:

dεpa, tq “

ÿ

j

w εj pa, tqµ

εj pa, tq, w ε

j pa, tq “g εj pa, tq

g εpa, tq(1)

d depends non-linearly on the population inputs: g 0j , µj , and bj .

§ Even with time-independent rates µεj pa, �Atq

ñ the aggregate death rate dεpa, tq depends on time, due to changes

of composition of the heterogeneous population.

13

Numerical results

§ Two applications

1 Impact of the age-pyramid heterogeneity.ë Compare order of magnitude of mortality changes induced by

compositional changes to constant mortality improvements.

2 Cause specific mortality reduction vs “reverse” cohort effect.

ë Compensation of cause-specific mortality reduction due to adverse

compositional changes in some cohorts.

§ We consider a synthetic population composed of the most and least

deprived IMD quintile (illustrative purpose).

14

Three demographic scenarios

A Scenario A: Population evolution with time-invariant mortality

Compositional changes isolated ñ death rates in each subpopulation

do not depend on time:

dεpa, tq “ µε1paqwε1 pa, tq ` µ

ε5paqw

ε5 pa, tq.

B Scenario B: Population evolution with mortality improvement

Constant annual mortality improvement rates of r “ 0.5%:

dεpa, tq “ µε1paqp1´ rqtw ε1 pa, tq ` µ

ε5paqp1´ rqtw ε

5 pa, tq.

C Scenario C: Mortality improvements without composition changes

dεpa, tq “ µε1paqp1´ rqtw ε1 paq ` µ

ε5paqp1´ rqtw ε

5 paq.

Mortality rates and initial age-pyramid fitted to the data for year 1981 and

2015.

15

Mortality improvement rates (males)

Figure: Average annual mortality improvement rates over years 0-30

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−0.5%

−0.4%

−0.3%

−0.2%

−0.1%

0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

65 70 75 80 85

Age

Ave

rage

d an

nual

impr

ovem

ent r

ates

Scenario ● ● ●A B C

(a) 1981 Inputs

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−0.5%

−0.4%

−0.3%

−0.2%

−0.1%

0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

65 70 75 80 85

Age

Ave

rage

d an

nual

impr

ovem

ent r

ates

Scenario ● ● ●A B C

(b) 2015 Inputs

16

Summary

§ 1981 initial population: positive contribution of changes in the

composition of the 65+ age class.

§ 2015 initial population: negative contribution of composition changes

ñ might offset future mortality improvement rates.

§ Order of magnitude of age-pyramid heterogeneity impact in annual

mortality improvement rates of 0.2%- 0.5%.

17

Composition changes and compensation effect

Example of scenario illustrating impact of changes of demographic rates:

§ Cause specific reduction of mortality vs “reverse” cohort effect (adverse

composition changes quantified by changes in birth patterns).

§ Baseline (“neutral”) scenario:

Death rates: time-invariant death rates in each population (year 2015).

Birth rates same birth rates in each population (England, 2015).

Initial pyramid our aim is to limit influence of initial pyramids

evolution: (((((((

((hhhhhhhhhSame initial pyramid ñ Stable pyramids.

18

Scenario 1a: cause of death reduction

Figure: Reduction of mortality rates from cardiovascular disease (CVD)

Males

0 10 20 30 40 50 60 70 80 90 10053.4

53.6

53.8

54

54.2

54.4

54.6

54.8

55

55.2

Time

Life

exp

ecta

ncy

at a

ge 2

5

Baseline scenario

Scenario 1a cause 2 (0%,−10%)



(a) Male period life expectancy at 25

Males

0 10 20 30 40 50 60 70 80 90 10053.4

53.6

53.8

54

54.2

54.4

54.6

54.8

55

55.2

Time

Life

exp

ecta

ncy

at a

ge 2

5

Baseline scenario




(b) Male cohort life expectancy at 25

§ Reduction of CoD mortality from CVD (10, 20 and 30%) over a period

of 30 years, starting at t “ 40.19

Scenario 1b: “Reverse” Cohort effect

Males

0 10 20 30 40 50 60 70 80 90 10053.4

53.6

53.8

54

54.2

54.4

54.6

54.8

55

55.2

Time

Life

exp

ecta

ncy

at a

ge 2

5

Baseline scenario

Scenario 1b (20%,0%)



(a) Male period life expectancy at 25

Males

0 10 20 30 40 50 60 70 80 90 10053.4

53.6

53.8

54

54.2

54.4

54.6

54.8

55

55.2

TimeLi

fe e

xpec

tanc

y at

age

25

Baseline scenario




(b) Male cohort life expectancy at 25

§ Reverse cohort effect: increase of birth rates in most deprived subgroup

over period r0, 20s.

§ Õ 60% ñ cohorts composed of 63% of most deprived subgroup.

20

Scenario 2: combined CoD reduction and cohort effect

Males

0 10 20 30 40 50 60 70 80 90 10053.4

53.6

53.8

54

54.2

54.4

54.6

54.8

55

55.2

Time

Life

exp

ecta

ncy

at a

ge 2

5

Baseline scenario


Scenario 2 cause 2 (20%,−20%)



Figure: Male period life expectancy at 25

When the population heterogeneity is not taken into account, cause-of-death

mortality reduction could be compensated for and/or misinterpreted depending on

the population composition evolution.

21

Outline

1 Introduction




22

The consistency issue

§ Population divided in p risk classes.

Example: national population divided in socioeconomic or geographical

subgroups...

§ Many forecasts are made simultaneously at the national level and at

the subgroup level.

§ Issue Are forecasts consistent?

(Aggregation relation) µxt “

pÿ

j“1

µjxtW

jxt ?

§ Problem when comparing policies, question robustness of forecasts.

23

Existing approaches

§ Relative approach (e.g.Jarner and Kryger (2011), Cairns et al. (2011),

Villegas and Haberman (2014), Li et al. (2015))

Method: First forecast of the national population and then mortality

differentials µjxt ´ µxt .

Advantages: national mortality data available for longer period, more

precise age-groups, estimations less volatile.

Drawback: No consistency!

§ Bottom-up approach:

Method: First forecast subnational mortality and then pose

µxt “řp

j“1 µjxtW

jxt .

Advantages: consistent forecasts.

Drawbacks: poor/missing data quality, small datasets.

§ Optimal combination : Combines the advantages of the two

approaches.

24

Optimal combination

§ Forecast reconciliation method used in economics, operation research

and recently for mortality rates (Shang and Hyndman (2017), Shang and

Haberman (2017)).

§ Vector notations: βnpxq “

¨

˚

˚

˝

µ1n

...

µpn

˛

‹

‹

‚

pxq, Ytpxq “

˜

µt

βt

¸

pxq.

§ Consistency equation:

Yn “Wnβn, Wn “

¨

˚

˚

˚

˚

˚

˝

W 1n W 2

n ¨ ¨ ¨ W pn

1 0 ¨ ¨ ¨ 0...

. . . 0

0 ¨ ¨ ¨ 1

˛

‹

‹

‹

‹

‹

‚

.

25

Optimal combination

Method:

1 First forecast mortality rates pYnqn“t..T using any method:

Yn ‰Wnβn.

2 Revise the subpopulations forecasts βn “tpµ1

n, ¨ ¨ ¨ , µpnq so that the

forecast Yn “Wnβn obtained by bottom-up is as close as possible as

the initial forecast Y .

Yn “Wnβn ` εn.

3 Replace Y by the consistent forecast Yn “Wnβn.

How can the relative exposures W jnpxq “

E jnpxq

Enpxqbe forecasted?

26

Population composition forecast

Proposed methods

§ Forecast Exposure to Risk at each age x pE jnpxqqn“t..T using

independent time series.

§ Constant Exposure to risk E jt`hpxq “ E j

t px ´ hq.

Reseach program

§ Population dynamics model provides a natural framework for the

projection of relative exposures, using all the age-pyramid information.

§ Adapt the bottom-up and optimal combination method in this context.

§ Challenge:

Yn “Wnppβsqs“t...nqβn,

27

Conclusion and perspectives

Population dynamics framework ñ study of interactions between

changes in socioeconomic composition and mortality indicators:

§ Evidence from data: future evolution of the population socioeconomic

composition could have a different impact on mortality than 30y ago.

§ Cause-of-death mortality changes can be compensated and/or

misinterpreted in the presence of heterogeneity.

Perspectives

§ Take into account internal migrations: ñ non-linear mortality models.

§ Stochastic modelling

ë R package IBMPopSim in preparation,(with D. Giorgi and V. Lemaire).

§ Consistency of mortality forecasts under different demographic

scenarios.

28

Thank you for you attention!

29

Initial age-pyramids

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100+

19811976

19711966

19611956

19511946

1941

19361931

19261921

19161911

19061901

18961891

18861881

150 100 50 0 50 100 150Number of individuals (thousands)

Age

Year of birth

Type Males Females IMD Quintile 1 5

(a) 1981 Inputs

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100+

20152010

20052000

19951990

19851980

1975

19701965

19601955

19501945

19401935

19301925

19201915

150 100 50 0 50 100 150Number of individuals (thousands)

Age

Year of birth

Type Males Females IMD Quintile 1 5

(b) 2015 Inputs

30

Period life expectancy

Figure: Evolution of male life expectancy at age 65

65 60 55 50 45 40 35 30 25

13.0

13.1

13.2

13.3

13.4

13.5

13.6

13.7

13.8

13.9

14.0

14.1

14.2

14.3

14.4

14.5

14.6

14.7

0 5 10 15 20 25 30 35 40

Age at initial time

Time

Per

iod

life

expe

ctan

cy

Scenario A B C

(a) 1981 Inputs

65 60 55 50 45 40 35 30 25

17.9

18.0

18.1

18.2

18.3

18.4

18.5

18.6

18.7

18.8

18.9

19.0

19.1

19.2

19.3

19.4

19.5

19.6

19.7

19.8

19.9

0 5 10 15 20 25 30 35 40

Age at initial time

Time

Per

iod

life

expe

ctan

cy

Scenario A B C

(b) 2015 Inputs

§ Life expectancy at t “ 30: surviving individuals initially 35+31

Birth-Death-Swap processes

Pathwise construction of Birth-Death-Swap systems leading to an averaging result in presence of two

timescales, with N. El Karoui

§ Goal: To study the random evolution of an heterogeneous population

including:

A time-varying random environment.

Model changes in the population’s composition induced by interacting

individuals changing characteristics.

§ Main contributions:

General mathematical framework and tools to study such processes.

Study of the aggregated “macro” dynamic produced by such models.

§ Averaging result: aggregated mortality rates are approximated by

“averaged” rates depending non-trivially on the number of individuals

in the population.

32

Interpreting mortality trends in the presence of heterogeneity ......Modeling heterogeneous...

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