Smoothing mortality rates

November 4, 2013
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This morning, I was working with Julie, a student of mine, coming from Rennes, on mortality tables. Actually, we work on genealogical datasets from a small region in Québec, and we can observe a lot of volatiliy. If I borrow one of her graph, we get something like

Since we have some missing data, we wanted to use some Generalized Nonlinear Models. So let us see how to get a smooth estimator of the mortality surface.  We will write some code that we can use on our data later on (the dataset we have has been obtained after signing a lot of official documents, and I guess I cannot upload it here, even partially).

DEATH <- read.table(
"http://freakonometrics.free.fr/Deces-France.txt",
header=TRUE)
EXPO  <- read.table(
"http://freakonometrics.free.fr/Exposures-France.txt",
header=TRUE,skip=2)
library(gnm)
D=DEATH$Male
E=EXPO$Male
A=as.numeric(as.character(DEATH$Age))
Y=DEATH$Year
I=(A<100)
base=data.frame(D=D,E=E,Y=Y,A=A)
subbase=base[I,]
subbase=subbase[!is.na(subbase$A),]

The first idea can be to use a Poisson model, where the mortality rate is a smooth function of the age and the year, something like

that can be estimated using

library(mgcv)
regbsp=gam(D~s(A,Y,bs="cr")+offset(log(E)),data=subbase,family=quasipoisson)
predmodel=function(a,y) predict(regbsp,newdata=data.frame(A=a,Y=y,E=1))
vX=trunc(seq(0,99,length=41))
vY=trunc(seq(1900,2005,length=41))
vZ=outer(vX,vY,predmodel)
persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)",
ylab="Years (1900-2005)",zlab="Mortality rate (log)")

The mortality surface is here

It is also possible to extract the average value of the years, which is the interpretation of the  coefficient in the Lee-Carter model,

predAx=function(a) mean(predict(regbsp,newdata=data.frame(A=a,
Y=seq(min(subbase$Y),max(subbase$Y)),E=1)))
plot(seq(0,99),Vectorize(predAx)(seq(0,99)),col="red",lwd=3,type="l")

We have the following smoothed mortality rate

Recall that the Lee-Carter model is

where parameter estimates can be obtained using

regnp=gnm(D~factor(A)+Mult(factor(A),factor(Y))+offset(log(E)),
data=subbase,family=quasipoisson)
predmodel=function(a,y) predict(regnp,newdata=data.frame(A=a,Y=y,E=1))
vZ=outer(vX,vY,predmodel)
persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)",
ylab="Years (1900-2005)",zlab="Mortality rate (log)")

The (crude) mortality surface is

with the following  coefficients.

plot(seq(1,99),coefficients(regnp)[2:100],col="red",lwd=3,type="l")

Here we have a lot of coefficients, and unfortunately, on a smaller dataset, we have much more variability. Can we smooth our Lee-Carter model ? To get something which looks like

Actually, we can, and the code is rather simple

library(splines)
knotsA=c(20,40,60,80)
knotsY=c(1920,1945,1980,2000)
regsp=gnm(D~bs(subbase$A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3)+
Mult(bs(subbase$A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3),
 bs(subbase$Y,knots=knotsY,Boundary.knots=range(subbase$Y),degre=3))+
offset(log(E)),data=subbase, family=quasipoisson) 
BpA=bs(seq(0,99),knots=knotsA,Boundary.knots=range(subbase$A),degre=3) 
BpY=bs(seq(min(subbase$Y),max(subbase$Y)),knots=knotsY,Boundary.knots= range(subbase$Y),degre=3) 
predmodel=function(a,y) 
predict(regsp,newdata=data.frame(A=a,Y=y,E=1)) v
Z=outer(vX,vY,predmodel) 
persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)", 
ylab="Years (1900-2005)",zlab="Mortality rate (log)")

The mortality surface is now

and again, it is possible to extract the average mortality rate, as a function of the age, over the years,

BpA=bs(seq(0,99),knots=knotsA,Boundary.knots=range(subbase$A),degre=3)
Ax=BpA%*%coefficients(regsp)[2:8]
plot(seq(0,99),Ax,col="red",lwd=3,type="l")

We can then play with the smoothing parameters of the spline functions, and see the impact on the mortality surface

knotsA=seq(5,95,by=5)
knotsY=seq(1910,2000,by=10)
regsp=gnm(D~bs(A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3)+
Mult(bs(A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3),
bs(Y,knots=knotsY,Boundary.knots=range(subbase$Y),degre=3))
+offset(log(E)),data=subbase,family=quasipoisson)
predmodel=function(a,y) predict(regsp,newdata=data.frame(A=a,Y=y,E=1))
vZ=outer(vX,vY,predmodel)
persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)",
ylab="Years (1900-2005)",zlab="Mortality rate (log)")

We now have to use those functions our our small data sample ! That should be fun….

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