,
Living to 100: Mortality Modelling
Modelling, Measurement and Management ofLongevity Risk
Andrew J.G. Cairns
Heriot-Watt University, Edinburgh
Director, Actuarial Research Centre, IFoA
Society of Actuaries Annual Meeting, Boston, October 2017
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The Actuarial Research Centre (ARC)A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty ofActuaries’ (IFoA) network of actuarial researchers around the world.The ARC seeks to deliver cutting-edge research programmes that addresssome of the significant, global challenges in actuarial science, through apartnership of the actuarial profession, the academic community andpractitioners.The ’Modelling, Measurement and Management of Longevity andMorbidity Risk’ research programme is being funded by the ARC, theSoA and the CIA.
www.actuaries.org.uk/arc
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ARC research program themes
Improved models for mortality
Key drivers of mortality
Management of longevity risk
Morbidity risk modelling for critical illnessinsurance
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Outline
Part 1: All cause mortality modelling
Introduction to stochastic mortality modelsWhy?Example applications
Part 2: Key drivers
Education levelCause of deathHealth inequalities
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Part 1: All Cause Mortality Modelling
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US Historical Death Rates
1940 1960 1980 2000 2020 2040
0.00
100.
0025
?
Year
Dea
th R
ate
(log
scal
e)
US Males Aged 20
1940 1960 1980 2000 2020 2040
0.00
20.
004
0.00
8
?
Year
Dea
th R
ate
(log
scal
e)
US Males Aged 40
1940 1960 1980 2000 2020 2040
0.01
00.
020
0.04
0
?
Year
Dea
th R
ate
(log
scal
e)
US Males Aged 60
1940 1960 1980 2000 2020 2040
0.05
0.10
0.20
?
Year
Dea
th R
ate
(log
scal
e)
US Males Aged 80
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Graphical Diagnostics
Mortality is falling
Different improvement rates at different ages
Different improvement rates over differentperiodsImprovements are random
Short term fluctuationsLong term trends
All stylised factsOther countries:
Some similaritiesSome different patterns
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Why do we need stochastic mortality models?
Data ⇒ future mortality is uncertain
Good risk management
Setting risk reserves
Regulatory capital requirements (e.g. SolvencyII)
Life insurance contracts with embedded options
Pricing and hedging mortality-linked securities
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Modelling
Aims:to develop the best models for forecasting futureuncertain mortality;
general desirable criteriacomplexity of model ↔ complexity ofproblem;longevity versus brevity risk;
measurement of risk;
valuation of future risky cashflows.
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Management
Aims:active management of mortality and longevityrisk;
internal (e.g. product design; naturalhedging)over-the-counter deals (OTC)securitisation
part of overall package of good riskmanagement.
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Stochastic Mortality Models
Two basic examples:
Lee-Carter Model (1992)
Cairns-Blake-Dowd Model (CBD) (2006)
Stochastic model:
Central forecast
Uncertainty around the central forecast
Good ERM ⇒ Use a combination of stochasticprojections plus some deterministic scenarios orstress tests
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The Lee-Carter Model
Death rate:
m(t, x) =D(t, x)
E (t, x)=
deaths(t, x)
average population(t, x)
Year t; Age x .
LC: logm(t, x) = α(x) + β(x)κ(t)
α(x) = base table; age effect
β(x) = age effect
κ(t) = period effect
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The Lee-Carter Model
logm(t, x) = α(x) + β(x)κ(t)
Estimate α(x), β(x), κ(t) from historical data“Traditional” model:
Fit a random walk model to historical κ(t)Simulate future scenarios for κ(t)Calculate future mortality scenarios givenκ(t)
Alternative models for κ(t) can be used
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The CBD Model
q(t, x) = Probability of death in year t giveninitially exact age x .
q(t, x) ≈ 1− exp[−m(t, x)]
logit q(t, x) = log
(q
1− q
)= κ1(t) + κ2(t)(x − x̄)
κ1(t) = period effect; affects level
κ2(t) = period effect; affects slope
x̄ = mean age
Captures big picture at higher ages
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Comparison
LC ⇒ all mortality rates dependent on a single κ(t)⇒ rates at all ages perfectly correlated
CBD ⇒ simpler age effects (1 and x − x̄)but two period effects⇒ richer correlation structure
CBD linearity ⇒not good for younger ages
Historical data:Different improvements at different ages over differenttime periods⇒ need more than one period effect
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Applications: Scenario Generation
Example: the Lee Carter Model
(Applied to a synthetic dataset)
logm(t, x) = α(x) + β(x)κ(t)
Choose a time series model for κ(t)
Calibrate the time series parameters using dataup to the current time (time 0)
Generate j = 1, . . . ,N stochastic scenarios ofκ(t)
κ1(t), . . . , κN(t)
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Applications: Scenario Generation
Generate N scenarios for the future m(t, x)mj(t, x) for j = 1, . . . ,N , t = 0, 1, 2, . . .,x = x0, . . . , x1
Generate N scenarios for the survivor index,Sj(t, x)
Calculate financial functions
+ variations for some financial applications.
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Applications: Scenario Generation, κ(t)
−30 −20 −10 0 10 20 30
−1.
0−
0.5
0.0
0.5
Historical Simulated
Period Effect: One Scenario
Time
Per
iod
Effe
ct, k
appa
(t)
κ(t): Generate scenario 1
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Applications: Scenario Generation, κ(t)
−30 −20 −10 0 10 20 30
−1.
0−
0.5
0.0
0.5
Historical Simulated
Period Effect: Multiple Scenarios
Time
Per
iod
Effe
ct, k
appa
(t)
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Applications: Scenario Generation, κ(t)
−30 −20 −10 0 10 20 30
−1.
0−
0.5
0.0
0.5
Historical Simulated
Period Effect: Fan Chart
Time
Per
iod
Effe
ct, k
appa
(t)
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Applications: Scenario Generation, Future m(t, x)
0 5 10 15 20 25 30
0.00
60.
008
0.01
20.
016
Death Rates, Age 65: One Scenario
Time
Dea
th R
ate
(log
scal
e)
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Applications: Scenario Generation, Future m(t, x)
0 5 10 15 20 25 30
0.00
60.
008
0.01
20.
016
Death Rates, Age 65: Multiple Scenarios
Time
Dea
th R
ate
(log
scal
e)
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Applications: Scenario Generation, Future m(t, x)
0 5 10 15 20 25 30
0.00
60.
008
0.01
20.
016
Death Rates, Age 65: Fan Chart
Time
Dea
th R
ate
(log
scal
e)
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Annuity Pricing Requires Cohort Rates
0 5 10 15 20 25 30
6570
7580
8590
Extract Cohort Death Rates, m(t,x+t−1)
Time
Age
Annuity valuation ⇒ follow cohorts
m(0, x)→ m(1, x + 1)→ m(2, x + 2) . . .
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Annuity Pricing Requires Cohort Rates
65 70 75 80 85 90 95 100
0.01
0.02
0.05
0.10
0.20
Cohort Death Rates From Age 65:One Scenario
Cohort Age
Dea
th R
ate
(log
scal
e)
Annuity valuation ⇒ follow cohorts
m(0, x)→ m(1, x + 1)→ m(2, x + 2) . . .
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Annuity Pricing Requires Cohort Rates
65 70 75 80 85 90 95 100
0.01
0.02
0.05
0.10
0.20
Cohort Death Rates From Age 65:Multpiple Scenarios
Cohort Age
Dea
th R
ate
(log
scal
e)
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Annuity Pricing Requires Cohort Rates
65 70 75 80 85 90 95 100
0.01
0.02
0.05
0.10
0.20
Cohort Death Rates From Age 65:Fan Chart
Cohort Age
Dea
th R
ate
(log
scal
e)
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Cohort Survivor Index
65 70 75 80 85 90 95 100
0.0
0.2
0.4
0.6
0.8
1.0
Survivorship From Age 65:One Scenario
Cohort Age
Sur
vivo
r In
dex
(log
scal
e)
Cohort death rates −→ cohort survivorship
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Cohort Survivor Index
65 70 75 80 85 90 95 100
0.0
0.2
0.4
0.6
0.8
1.0
Survivorship From Age 65:Multiple Scenarios
Cohort Age
Sur
vivo
r In
dex
(log
scal
e)
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Cohort Survivor Index
65 70 75 80 85 90 95 100
0.0
0.2
0.4
0.6
0.8
1.0
Survivorship From Age 65:Fan Chart
Cohort Age
Sur
vivo
r In
dex
(log
scal
e)
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Life Expectancy
17 18 19 20 21 22
010
020
030
040
050
0
Cohort Life Expectancy from Age 65
Life Expectancy From Age 65
Fre
quen
cy
17 18 19 20 21 22
0.0
0.2
0.4
0.6
0.8
1.0
Cohort Life Expectancy from Age 65
Life Expectancy From Age 65
Cum
ulat
ive
Pro
babi
lity
Cohort survivorship −→ ex post cohort life expectancyEquivalent to a continuous annuity with 0% interest
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Annuity Reserving
14 15 16 17
020
040
060
080
0
Present Value of Annuity from Age 65
Present Value ofAnnuity From Age 65
Fre
quen
cy
13.5 14.5 15.5 16.5
0.0
0.2
0.4
0.6
0.8
1.0
Present Value of Annuity from Age 65
Present Value ofAnnuity From Age 65
Cum
ulat
ive
Pro
babi
lity
Annuity of 1 per annum payable annually in arrears
Interest rate: 2%
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A Real Example: US Male Period Life Expectancy
0 5 10 15 20 25 30
6570
7580
8590
Extract Period Death Rates, m(t,x+t−1)
Time
Age
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A Real Example: US Male Period Life Expectancy
1990 2000 2010 2020 2030 2040 2050
1416
1820
2224
Year
Life
Exp
ecta
ncy
Fro
m A
ge 6
5
US Male Period Life Expectancy From Age 65(Stochastic Model: CBD−X−K3−G)
Mortality improvement rate ≈ 1.7% p.a. at ages 65-85.
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How to incorporate Expert Judgement?
E.g. CBD model ⇒mj
CBD(t, x) scenariosm̄CBD(t, x) central forecast
Expert judgement ⇒m̂(t, x) (central) forecast
Blending ⇒ stochastic scenario j becomes
mj(t, x) =mj
CBD(t, x)
m̄CBD(t, x)× m̂(t, x)
Fully stochastic ⇒ full risk assessment
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How to incorporate Expert Judgement?
A variation on this is required by UK lifeinsurance regulators
⇒ Don’t ignore stochastic models simplybecause you disagree with the central forecast!
Additionally: new approaches to bring the twotogether are being developed
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Part 2: Key Drivers
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Drill into the Detail of US Data
Level of educational attainment ⇒ predictor
Individual cause of death ⇒ outcome
Beware of grade inflation
Help to understand trends in national data andsubpopulations (e.g. white collar pension plan)
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Data Sources
Total Exposures: Human Mortality Database(smoothed to mitigate anomalies)
CDC deaths: cause of death + education(+ ethnic group)
CPS survey data: education proportions
Research ⇒smart synthesis of three data sources
improved, less noisy, exposures by education level
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Purpose of looking at cause of death data
What are the key drivers of all-cause mortality?
How are the key drivers changing over time?Which causes of death have high levels of inequality:
by educationother predictors
Insight into mortality underpinning life insurance andpensions
Insight into potential future mortality improvements
Beware of
changes in ICD classification of deaths (e.g. 1999)drift in how deaths are classifiedchanging education levels (grade inflation)
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Education Levels
EducationLow education Primary and lower secondary educationMedium education Upper secondary educationHigh education Tertiary education
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Cause of Death Groupings
1 Infectious diseases incl. tuberculosis 2 Cancer: mouth, gullet, stomach3 Cancer: gut, rectum 4 Cancer: lung, larynx, ..5 Cancer: breast 6 Cancer: uterus, cervix7 Cancer: prostate, testicular 8 Cancer: bones, skin9 Cancer: lymphatic, blood-forming tissue 10 Benign tumours11 Diseases: blood 12 Diabetes13 Mental illness 14 Meningitis + nervous system (Alzh.)15 Blood pressure + rheumatic fever 16 Ischaemic heart diseases17 Other heart diseases 18 Diseases: cerebrovascular19 Diseases: circulatory 20 Diseases: lungs, breathing21 Diseases: digestive 22 Diseases: urine, kidney,...23 Diseases: skin, bone, tissue 24 Senility without mental illness25 Road/other accidents 26 Other causes27 Alcohol → liver disease 28 Suicide29 Accidental Poisonings
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US Education Data
Males and Females (2)
Single ages 55-75 (21)
Single years 1989-2015 (27)
Causes of death (29)
Low, medium & high education level (3)
Note: HMD’s Human Cause of Death Database ⇒All ages (5’s), 1999-2015, No education
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US Education Data: Growing Inequality, Males
55 60 65 70 75
0.00
20.
005
0.01
00.
020
0.05
00.
100
1989
2015
1989
2015
High
Low
Low Ed 1989Low Ed 2002Low Ed 2015High Ed 1989High Ed 2002High Ed 2015
Age
Dea
th R
ate
(log
scal
e)Male All Cause Death Rates by Education Group
For 1989 and 2015
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US Education Data: Growing Inequality, Females
55 60 65 70 75
0.00
20.
005
0.01
00.
020
0.05
0
19892015
1989
2015
High
Low
Low Ed 1989Low Ed 2002Low Ed 2015High Ed 1989High Ed 2002High Ed 2015
Age
Dea
th R
ate
(log
scal
e)Female All Cause Death Rates by Education Group
For 1989 and 2015
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Proportion of Males with Low Education
US Males 1989−2015 Ages 55−75:Proportion of Population with Low Education
Year 1989−2015
Age
555657585960616263646566676869707172737475
1990 1994 1998 2002 2006 2010 2014
30
40
50
60
70
80
Cohort diagonals ⇒ falling percentage
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US Education Data: CoD Death Rates
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2000 Ischaemic heart diseases
Age
CoD
Dea
th R
ate
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2015 Ischaemic heart diseases
Age
CoD
Dea
th R
ate
Widening gap
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US Education Data: CoD Death Rates
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2000 Cancer: lung, larynx, ..
Age
CoD
Dea
th R
ate
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2015 Cancer: lung, larynx, ..
Age
CoD
Dea
th R
ate
Widening gap
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US Education Data: CoD Death Rates
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2000 Diabetes
Age
CoD
Dea
th R
ate
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2015 Diabetes
Age
CoD
Dea
th R
ate
Widening gap; Mixed improvements
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US Education Data: CoD Death Rates
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2000 Meningitis + nervous system (Alzh.)
Age
CoD
Dea
th R
ate
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2015 Meningitis + nervous system (Alzh.)
Age
CoD
Dea
th R
ate
Widening gap; almost no improvements
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US Education Data: CoD Death Rates
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2000 Accidental Poisonings
Age
CoD
Dea
th R
ate
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2015 Accidental Poisonings
Age
CoD
Dea
th R
ate
Case & Deaton (2015) ⇒ Accidental poisoning ↗
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US Education Data: CoD Death Rates
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2000 Alcohol −> liver
Age
CoD
Dea
th R
ate
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2015 Alcohol −> liver
Age
CoD
Dea
th R
ate
Widening gap
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US Education Data: CoD Death Rates
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2000 Cancer: prostate, testicular
Age
CoD
Dea
th R
ate
55 60 65 70 75
1e−
051e
−04
1e−
031e
−02
1e−
01 Low EduMediumHigh
Year 2015 Cancer: prostate, testicular
Age
CoD
Dea
th R
ate
Denmark ⇒ almost NO gap by education;Denmark ⇒ small gap by affluence; smaller than US by education
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Cause of Death Data: Health Inequalities
Some causes of death have no obvious link tolifestyle/affluence/educatione.g. Prostate CancerCancerUK:Prostate cancer is not clearly linked to anypreventable risk factors.
But education level ⇒ inequalities
Possible explanations (a very non-expert view)
onset is not dependent on lifestyle/affluence/educationBUT lower educated ⇒
??? poorer health insurance coverage??? later diagnosis??? engage less well with treatment process??? lower quality housing/diet etc.
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US Males: Low versus High Education
Do Low and High education groups have the sameCoD rate?
Four × 5-year age groups
29 causes of death
Signs Test (count low edu. > high edu. mort.)
29× 4 = 116 individual tests
Reject equality hypothesis in all but one test
Accept H0 (p = 0.08) for only one pairing:Meningitis + nervous system (Alzh.), 70-74
Most p-values < 10−6
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Summary
Future work
Analysis of sub-national datasetse.g. SoA Group and Individual Annuity datae.g. individual pension plan dataMultiple population modelling
E: [email protected] W: www.macs.hw.ac.uk/∼andrewc
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Thank You!
Questions?
E: [email protected] W: www.macs.hw.ac.uk/∼andrewc
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Discussion Point
Medicare kicks in after age 65
But no obvious impact on inequality gap
Although inequality gap naturally narrows withage
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CoD Death Rates: Different Shapes & Patterns
40 50 60 70 80 90
1e−
051e
−04
1e−
031e
−02
1e−
01Infectious diseases incl. tuberculosis
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−04
1e−
031e
−02
1e−
01
Meningitis + nervous system (Alzh.)
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−04
1e−
031e
−02
1e−
01
Ischaemic heart diseases
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−04
1e−
031e
−02
1e−
01
Diseases: circulatory
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−04
1e−
031e
−02
1e−
01Diseases: lungs, breathing
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−04
1e−
031e
−02
1e−
01
Diseases: urine, kidney,...
Dea
th R
ate
(log
scal
e)
12345678910
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CoD Death Rates: Different Shapes & Patterns
40 50 60 70 80 90
1e−
051e
−03
1e−
01
Cancer: gut, rectum
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−03
1e−
01
Cancer: lung, larynx, ..
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−03
1e−
01
Cancer: prostate, testicular
Dea
th R
ate
(log
scal
e)
12345678910
40 50 60 70 80 90
1e−
051e
−03
1e−
01
Cancer: bones, skin
Dea
th R
ate
(log
scal
e)
12345678910
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Shapes: Conclusions
Typically:Non-cancerous diseases ⇒ approximately exponentialgrowthNeoplasms (cancers) ⇒ subexponential ???polynomial
What does this reveal about different diseasemechanisms?
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