 Reinterpreting Lee-Carter Mortality Model

Last week, while I was giving my crash course on R for insurance, we’ve been discussing possible extensions of Lee & Carter (1992) model. If we look at the seminal paper, the model is defined as follows Hence, it means that This would be a (non)linear model on the logarithm of the mortality rate. A non-equivalent, but alternative expression might be

which could be obtained as a Gaussian model, with a log-link function Actually, this model can be compared to the more popular one, introduced in 2002 by Natacha Brouhns, Michel Denuit and Jeroen Vermunt, where a Poisson regression is used to count deaths (with the exposure used as an offset variable) On our datasets

EXPO <- read.table(
"http://freakonometrics.free.fr/Exposures-France.txt",
"http://freakonometrics.free.fr/Deces-France.txt",
base=data.frame(
D=DEATH$Total, E=EXPO$Total,
X=as.factor(EXPO$Age), T=as.factor(EXPO$Year))
library(gnm)
listeage=c(101:109,"110+")
sousbase=base[! base$X %in% listeage,] # on met des nombres car il faut calculer T-X sousbase$X=as.numeric(as.character(sousbase$X)) sousbase$T=as.numeric(as.character(sousbase$T)) sousbase$C=sousbase$T-sousbase$X
sousbase$E=pmax(sousbase$E,sousbase$D) The codes to fit those models are the following LC.gauss <- gnm(D/E~ as.factor(X)+ Mult(as.factor(X),as.factor(T)), family=gaussian(link="log"), data=sousbase) LC.gauss.2 <- gnm(log(D/E)~ as.factor(X)+ Mult(as.factor(X),as.factor(T)), family=gaussian(link="identity"), data=sousbase) while for the Poisson regression is LC.poisson <- gnm(D~offset(log(E))+ as.factor(X)+ Mult(as.factor(X),as.factor(T)), family=poisson(link="log"), data=sousbase) To visualize the first component, the ‘s, use alphaG=coefficients(LC.gauss)+c(0, coefficients(LC.gauss)[2:101]) s=sd(residuals(LC.gauss.2)) alphaG2=coefficients(LC.gauss.2)+c(0, coefficients(LC.gauss.2)[2:101]) alphaGw=coefficients(LC.gauss.w)+c(0, coefficients(LC.gauss.w)[2:101]) We can then plot them plot(0:100,alphaP,col="black",type="l", xlab="Age") lines(0:100,alphaG,col="blue") legend(0,-1,c("Poisson","Gaussian"), lty=1,col=c("black","blue")) On small probabilities, the difference can be considered as substential. But for elderly, it seems that the difference is rather small. Now, the problem with a Poisson model is that it might generate a lot of deaths. Maybe more than the exposure actually. A natural idea is to consider a binomial model (which is a standard model in actuarial textbooks) The codes to run that (non)linear regression would be LC.binomiale <- gnm(D/E~ as.factor(X)+ Mult(as.factor(X),as.factor(T)), weights=E, family=binomial(link="logit"), data=sousbase) One more time, we can visualize the series of ‘s. alphaB=coefficients(LC.binomiale)+c(0, coefficients(LC.binomiale)[2:101]) Here, the difference is only on old people. For small probabilities, the binomial model can be approximated by a Poisson model. Which is what we observe. On elderly people, there is a large difference, and the Poisson model underestimates the probability of dying. Which makes sense, actually, since the number of deaths has to be smaller than the exposure. A Poisson model with a large parameter will have a (too) large variance. So the model will underestimate the probability. This is what we observe on the right. It is clearly a more realistic fit. 9 thoughts on “Reinterpreting Lee-Carter Mortality Model” 1. alfonso says: hi prof. i would like to know how in this code i have to do to change years of forecast, because of i have a similar data-set which deal about DEATH and EXPOSURE and i have to interpolate data and do forecasts with Lee-Carter model. 2. Arwa says: I want to apply Lee-Carter model to forecast mortality rate of lung cancer . The data I collectedfrom 2008 to 2014. but it has unequal intervals of age group and open last interval (0-4,5-14,15-24,25-44,45+). I will be grateful if you tell me how can I apply the Lee carter model on this data. 3. Turtje Rose says: hallo the LC.gauss.w var just appears without substantation, and in the text “invalid equation” also appears. Please explain. 1. Arthur Charpentier says: there was a typo in LaTeX… I need to fix it… should have been$D_{x,t}\sim\mathcal{B}\left(E_{x,t},\frac{e^{\alpha_x+\beta_x\cdot\kappa_t}}{1+e^{\alpha_x+\beta_x\cdot\kappa_t}}\right)\$

4. Fatin says:

Hi prof. First of all I would like to say that ur page is quite help me. Currently, I am a Master student at University in Malaysia. Im doing my thesis on study of Malaysia mortality rate by using poisson log bilinear model. Prof, can u help me? I would like to know what are the disadvantages of this model since my supervisor ask me to include it in my proposal. Thank you so much..

5. Tom says:

Hi, this might sound like a stupid question, but does EXPO mean births?

1. Arthur Charpentier says:

no, EXPO means alive at some age, and date

1. Tom says:

Thanks for the quick reply! BTW, how does your example compare to something like the ilc package?

1. Arthur Charpentier says:

I don’t know about ilc… I will look it it !

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