a history of mortality modelling from gompertz to lee

A History of Mortality Modellingfrom Gompertz to Lee-Carter

Everything in a single R package: MortalityLaws

Marius Pascariu and Vladimir Canudas-Romo

Max-Planck Odense Center on the Biodemography of AgingUniversity of Southern Denmark

8th Demographic Conference of "Young Demographers"

Prague, Czech Republic

February 16, 2017

Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 1 / 24

Agenda

Development of mortality modelling

Motivation

MortalityLaws R package

Discussion


What is Modelling?


Mortality Modelling Timeline

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DeMoivre

Gompertz

Makeham

Opperman

Thiele

Wittstein & Bumstead

Steffenson

Perks

Harper

Weibull

Van der Maen

Brillinger

Beard

Siler

Heligman−Pollard

Brooks et al.

Petrioli

Hartmann

Mode and Busby

Rogers and Planck

Martinelle

Kostaki

Carriere

Kannisto

Lee−Carter

Rogers and Little

1725 1750 1775 1800 1825 1850 1875 1900 1925 1950 1975 2000

YEAR


Age Pattern of Human Mortality

Female population in Czech Republic, 1970

Infant mortalityAccidental humpAdult mortalityOld-age mortality


Age Pattern of Human Mortality

Female population in Czech Republic, 1970

WeibullOppermanInverse-GompertzGompertzMakehamKannistoQuadratic—————-ThieleWittsteinHeligman-PollardCarriere...


Mortality Modelling & Forecasting in R

Demography (Hyndman 2014)Lee-Carter model and several of its variants

ilc (Butt, Haberman, and Shang 2014)Lee-Carter with cohorts and Lee-Carter under a Poisson framework

LifemetricsCBD and extensionsLee-Carter with cohorts and Lee-Carter under a Poisson framework

StMoMo Stochastic Mortality Modelling

ActuDistns Computes the probability density function for 44commonly used survival models


MortalityLaws: An R package for fitting human mortality

1 Smoothing data

2 Eliminating or/and reducing errors

3 Construct life tables

4 Facilitate comparisons of mortality improvement

5 Forecasting


MortalityLaws: An R package for fitting human mortality

REPOSITORY

Development version on Github:

https://github.com/mpascariu/MortalityLaws

INSTALLATIONMake sure you have the most recent version of R and

# install.packages(‘‘devtools’’)

library(devtools)

install github(‘‘mpascariu/MortalityLaws’’)

HELPAll functions are documented in the standard way, which means that onceyou load the package using library(MortalityLaws) you can just type?MortalityLaw to see the help file.


Mortality models already implemented in the package

Mortality laws Year Predictor

DeMoivre 1725 1(ω−x)

Gompertz 1825 aebx or 1σexp{

x−mσ

}Inverse-Gompertz 1

σexp{

x−mσ

}/

(exp

{e−(x−m)σ

}− 1

)

Makeham 1860 aebx + c

Inverse-Makeham 1σexp{

x−mσ

}/

(exp

{e−(x−m)σ

}− 1

)+ c

Opperman 1870 a√x+ b + c

√x

Thiele 1872 a1e−b1x + a2e

− 12b2(x−c)2

+ a3eb3x

Wittstein & Bumstead 1883 1ma−(mx)n + a−(M−x)n

Weibull 1939 1σ

(xm

)mσ−1

Inverse-Weibull 1σ

(xm

)−mσ−1/

(exp

{(xm

)−mσ

}− 1

)Siler 1979 a1e

−b1t + a2 + a3eb3t

Heligman - Pollard 1980 A(x+B)C + De−E(lnx −lnF )2 + GHx

Kannisto 1992 aebx

1+aebx+ c

Carriere 1992 s (x) = ψ1s1 (x) + ψ2s2 (x) + ψ3s3 (x)


Objective or loss functions

Find parameter estimates by minimizing any of the functions below:

Name Function

Poisson Log-Likelihood −∑

x {Dx logµ̂x − E cx µ̂x}+ c

Binomial Log-Likelihood −∑

x

{Dx log

[1− e−µ̂x

]− [E c

x − Dx ] µ̂x

}+ c

Loss Function 1 (LF1)(1− µ̂x

µx

)2LF2 log

(µ̂xµx

) 2

LF3 (µx−µ̂x )2

µx

LF4 (µx − µ̂x )2

LF5 (µx − µ̂x ) log(µxµ̂x

)LF6 |µx − µ̂x |


Model fitting using different objective functions

Heligman-Pollard applied to E&W 2010

mx = A(x+B)C + De−E(lnx −lnF )2 + GHx

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Age (x)

Log

Dea

th R

ate,

log(

mx)

0 20 40 60 80 1005e−

055e

−04

5e−

035e

−02

5e−

01

poissonLbinomialLLF1LF2LF5


1 The golden-section search - find the optimum by successivelynarrowing the range of values inside a interval. In R in stats Rpackage optimize function. Kiefer (1953)

2 Nelder-Mead method - approximates a local optimum of a problemwith n variables when the objective function varies smoothly and isunimodal. Implemented in stats R package, called in optim function.Nelder & Mead (1965)

3 PORT routines - provides unconstrained optimization andoptimization subject to box constraints for complicated functions. Seenlminb function, stats package.

4 Levenberg-Marquardt algorithm - damped least-squares method.Check nls.lm function in minpack.lm. Levenberg(1944);Marquardt(1963).


Model fitting: Data

Download data from HMD using ReadHMD(...) function


Model fitting: Data


Example: Siler Model - Czech Republic, 2012

Siler (1979): mx = a1e−b1t + a2 + a3e

b3t

Fit the model using: MortalityLaw(...)

Check output object: ls(fit.siler)



Summary : summary(fit.siler)



Generic plot function : plot(fit.siler)

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Observed and Fitted Age−Specific Death Rate

Age (x)

log(

mx)

0 25 50 75 100

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9−

8.2

−6.

1−

3.6

−0.

7

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Residual plot

Age (x)

Err

or

0 25 50 75 100

−0.

10−

0.05

0.00

Frequency0 20 40



Generic plot function : plot(fit.siler)

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Age (x)

log(

mx)

0 25 50 75 100

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9−

8.2

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1−

3.6

−0.

7

● ObservedFitted

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Residual plot

Age (x)

Err

or

0 25 50 75 100

−0.

10−

0.05

0.00

Frequency0 20 40

We have a problem!!!



Model fitted on the 0-75 age-range

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Age (x)

log(

mx)

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9−

8.2

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1−

3.6

−0.

7

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Residual plot

Age (x)

Err

or

0 18 37 56 75

−5e

−04

0e+

005e

−04

1e−

03

Frequency0 10 25


More mortality laws

Observed and fitted death rates between age 0 and 100 for female population inEngland & Wales

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−10.0

−7.5

−5.0

−2.5

0.0

0 25 50 75 100 0 25 50 75 100 0 25 50 75 100 0 25 50 75 100

Age (x)

Age

Spe

cific

−D

eath

Rat

e, m

x

Mortality Laws: Heligman−Pollard (1980) Thiele (1871) Wittstein (1883)


Old-age mortality

Observed and fitted old-age mortality for female population in England & Wales

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Fitted age−range

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Fitted age−range

1850 1900 1950 2013

0.00

0.25

0.50

0.75

1.00

80 90 100 110 120 80 90 100 110 120 80 90 100 110 120 80 90 100 110 120

Age (x)

Age

Spe

cific

−D

eath

Rat

e, m

x

Mortality Laws: Gompertz (1825) Kannisto (1992) Makeham (1867)


Download and install R package:

https://github.com/mpascariu/MortalityLaws

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Contact details:

[email protected]


a history of mortality modelling from gompertz to lee

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