A History of Mortality Modelling from Gompertz to Lee ...
Transcript of A History of Mortality Modelling from Gompertz to Lee ...
![Page 1: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/1.jpg)
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
![Page 2: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/2.jpg)
Agenda
Development of mortality modelling
Motivation
MortalityLaws R package
Discussion
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 2 / 24
![Page 3: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/3.jpg)
What is Modelling?
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 3 / 24
![Page 4: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/4.jpg)
What is Modelling?
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 4 / 24
![Page 5: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/5.jpg)
Mortality Modelling Timeline
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
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
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 5 / 24
![Page 6: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/6.jpg)
Age Pattern of Human Mortality
Female population in Czech Republic, 1970
Infant mortalityAccidental humpAdult mortalityOld-age mortality
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 6 / 24
![Page 7: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/7.jpg)
Age Pattern of Human Mortality
Female population in Czech Republic, 1970
WeibullOppermanInverse-GompertzGompertzMakehamKannistoQuadratic—————-ThieleWittsteinHeligman-PollardCarriere...
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 7 / 24
![Page 8: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/8.jpg)
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
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 8 / 24
![Page 9: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/9.jpg)
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
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 9 / 24
![Page 10: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/10.jpg)
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.
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 10 / 24
![Page 11: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/11.jpg)
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)
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 11 / 24
![Page 12: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/12.jpg)
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 |
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 12 / 24
![Page 13: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/13.jpg)
Model fitting using different objective functions
Heligman-Pollard applied to E&W 2010
mx = A(x+B)C + De−E(lnx −lnF )2 + GHx
●
●
●●●
●●●●
●●●●
●
●●●●●●●
●●
●●●●●●
●
●●
●●●●
●●●●
●●●●●●●●●●●
●●●
●●●●●●●●●
●●●●●●
●●●●●●●
●●
●●●
●●
●●●●●
●●●●
●●●●●●●●
●
Age (x)
Log
Dea
th R
ate,
log(
mx)
0 20 40 60 80 1005e−
055e
−04
5e−
035e
−02
5e−
01
poissonLbinomialLLF1LF2LF5
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 13 / 24
![Page 14: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/14.jpg)
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).
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 14 / 24
![Page 15: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/15.jpg)
Model fitting: Data
Download data from HMD using ReadHMD(...) function
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 15 / 24
![Page 16: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/16.jpg)
Model fitting: Data
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 16 / 24
![Page 17: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/17.jpg)
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)
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 17 / 24
![Page 18: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/18.jpg)
Example: Siler Model - Czech Republic, 2012
Summary : summary(fit.siler)
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 18 / 24
![Page 19: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/19.jpg)
Example: Siler Model - Czech Republic, 2012
Generic plot function : plot(fit.siler)
●
●
●
●
●
●●
●●●
●
●
●●●
●
●●●●●
●
●●●●●●●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●
Observed and Fitted Age−Specific Death Rate
Age (x)
log(
mx)
0 25 50 75 100
−9.
9−
8.2
−6.
1−
3.6
−0.
7
● ObservedFitted
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●●●
●
●
●●
●
●●
●
●●
Residual plot
Age (x)
Err
or
0 25 50 75 100
−0.
10−
0.05
0.00
Frequency0 20 40
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 19 / 24
![Page 20: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/20.jpg)
Example: Siler Model - Czech Republic, 2012
Generic plot function : plot(fit.siler)
●
●
●
●
●
●●
●●●
●
●
●●●
●
●●●●●
●
●●●●●●●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●
Observed and Fitted Age−Specific Death Rate
Age (x)
log(
mx)
0 25 50 75 100
−9.
9−
8.2
−6.
1−
3.6
−0.
7
● ObservedFitted
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●●●
●
●
●●
●
●●
●
●●
Residual plot
Age (x)
Err
or
0 25 50 75 100
−0.
10−
0.05
0.00
Frequency0 20 40
We have a problem!!!
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 20 / 24
![Page 21: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/21.jpg)
Example: Siler Model - Czech Republic, 2012
Model fitted on the 0-75 age-range
●
●
●
●
●
●●
●●●
●
●
●●●
●
●●●●●
●
●●●●●●●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●
Observed and Fitted Age−Specific Death Rate
Age (x)
log(
mx)
0 25 50 75 100
−9.
9−
8.2
−6.
1−
3.6
−0.
7
● ObservedFitted
●●●●●●●●●●
●●●●●
●
●
●●●●
●●●
●●●
●●
●●●
●
●●●●●●
●●●
●●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
Residual plot
Age (x)
Err
or
0 18 37 56 75
−5e
−04
0e+
005e
−04
1e−
03
Frequency0 10 25
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 21 / 24
![Page 22: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/22.jpg)
More mortality laws
Observed and fitted death rates between age 0 and 100 for female population inEngland & Wales
●
●
●●●●●●●●●●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●●●●
●●●●●●●●
●●●●●●●●
●●●●●●●
●●●●●●
●●●●●●
●●●●
●
●
●●●●●●●●●●●●●
●●●●●●●
●●●●●●●●●
●●●●●●●●●
●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●
●●●●●●
●●●●●●
●●●●●●●●
●●●●●●●●
●●●●●
●
●
●●●●●●●●●●
●●●
●●●●●●●●●●
●●●●●●●●●●
●●●●●●●●
●●●●●
●●●●●●●●●
●●●●●
●●●●●
●●●●●
●●●●●●●●●
●●●●●
●●●●●●
●●●●●●●●
●
●
●
●●●●●
●●●●●●●●●●●
●●●●●●●●
●●●●●●●●
●●●●●●●●●●
●●●●●●
●●●●●
●●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
1850 1900 1950 2013
−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)
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 22 / 24
![Page 23: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/23.jpg)
Old-age mortality
Observed and fitted old-age mortality for female population in England & Wales
●●●●●●●●●●●●●●●●
●●
●●
●●
●●
●●
●●
●
●
●
Fitted age−range
●●●●●●●●
●●●●●●●●
●●●●
●●
●●
●●
●
●
●
●Fitted age−range
●●●●●●●●●●●
●●●●
●●●●
●●●
●●
●
●●
●
●
●
●
●
Fitted age−range
●●●●●●●●●●●●●●●●●●
●●●
●●●
●●
●●
●●
●●●
●●
●
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)
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 23 / 24
![Page 24: A History of Mortality Modelling from Gompertz to Lee ...](https://reader034.fdocuments.net/reader034/viewer/2022051102/6276eb0d3f59175c09219c07/html5/thumbnails/24.jpg)
Download and install R package:
https://github.com/mpascariu/MortalityLaws
-
-
Contact details:
Pascariu & Canudas-Romo (MaxO) MortalityLaws February 16, 2017 24 / 24