# GLMs: link vs. distribution

Usually, when I give a course on GLMs, I try to insist on the fact that the link function is probably more important than the distribution. In order to illustrate, consider the following dataset, with 5 observations

x = c(1,2,3,4,5) y = c(1,2,4,2,6) base = data.frame(x,y)

Then consider several model, with various distributions, and either an identity link (and in that case $\mathbb{E}[Y|\mathbf{X}=\mathbf{x}]=\mathbf{x}^T\mathbf{\beta}$) or a log link function (so that $\mathbb{E}[Y|\mathbf{X}=\mathbf{x}]=e^{\mathbf{x}^T\mathbf{\beta}}$)

regNId = glm(y~x,family=gaussian(link="identity"),data=base) regNlog = glm(y~x,family=gaussian(link="log"),data=base) regPId = glm(y~x,family=poisson(link="identity"),data=base) regPlog = glm(y~x,family=poisson(link="log"),data=base) regGId = glm(y~x,family=Gamma(link="identity"),data=base) regGlog = glm(y~x,family=Gamma(link="log"),data=base) regIGId = glm(y~x,family=inverse.gaussian(link="identity"),data=base) regIGlog = glm(y~x,family=inverse.gaussian(link="log"),data=base

One can also consider some Tweedie distribution, to be even more general

library(statmod) regTwId = glm(y~x,family=tweedie(var.power=1.5,link.power=1),data=base) regTwlog = glm(y~x,family=tweedie(var.power=1.5,link.power=0),data=base)

Consider the prediction obtained in the first case, with the linear link function

library(RColorBrewer) darkcols = brewer.pal(8, "Dark2") plot(x,y,pch=19) abline(regNId,col=darkcols[1]) abline(regPId,col=darkcols[2]) abline(regGId,col=darkcols[3]) abline(regIGId,col=darkcols[4]) abline(regTwId,lty=2)

The predictions are very very close, aren’t they ? In the case of the exponential prediction, we obtain

plot(x,y,pch=19) u=seq(.8,5.2,by=.01) lines(u,predict(regNlog,newdata=data.frame(x=u),type="response"),col=darkcols[1]) lines(u,predict(regPlog,newdata=data.frame(x=u),type="response"),col=darkcols[2]) lines(u,predict(regGlog,newdata=data.frame(x=u),type="response"),col=darkcols[3]) lines(u,predict(regIGlog,newdata=data.frame(x=u),type="response"),col=darkcols[4]) lines(u,predict(regTwlog,newdata=data.frame(x=u),type="response"),lty=2)

We can actually look closer. For instance, in the linear case, consider the slope obtained with a Tweedie model (that will include all the parametric familes mentioned here, actually)

pente=function(gamma) summary(glm(y~x,family=tweedie(var.power=gamma,link.power=1),data=base))$coefficients[2,1:2] Vgamma = seq(-.5,3.5,by=.05) Vpente = Vectorize(pente)(Vgamma) plot(Vgamma,Vpente[1,],type="l",lwd=3,ylim=c(.965,1.03),xlab="power",ylab="slope") The slope here is always very very close to one ! Even more if we add a confidence interval plot(Vgamma,Vpente[1,]) lines(Vgamma,Vpente[1,]+1.96*Vpente[2,],lty=2) lines(Vgamma,Vpente[1,]-1.96*Vpente[2,],lty=2) Heuristically, for the Gamma regression, or the Inverse Gaussian one, because the variance is a power of the prediction, if the prediction is small (here on the left), the variance should be small. So, on the left of the graph, the error should be small with a higher power for the variance function. And that’s indeed what we observe here erreur=function(gamma) predict(glm(y~x,family=tweedie(var.power=gamma,link.power=1),data=base),newdata=data.frame(x=1),type="response")-y[x==1] Verreur = Vectorize(erreur)(Vgamma) plot(Vgamma,Verreur,type="l",lwd=3,ylim=c(-.1,.04),xlab="power",ylab="error") abline(h=0,lty=2) Of course, we can do the same with the exponential models pente=function(gamma) summary(glm(y~x,family=tweedie(var.power=gamma,link.power=0),data=base))$coefficients[2,1:2] Vpente = Vectorize(pente)(Vgamma) plot(Vgamma,Vpente[1,],type="l",lwd=3)

or, if we add the confidence bands, we obtain

plot(Vgamma,Vpente[1,],ylim=c(0,.8),type="l",lwd=3,xlab="power",ylab="slope") lines(Vgamma,Vpente[1,]+1.96*Vpente[2,],lty=2) lines(Vgamma,Vpente[1,]-1.96*Vpente[2,],lty=2)

So here also, the “slope” is rather similar… And if we look at the error we make on the left part of the graph, we obtain

erreur=function(gamma) predict(glm(y~x,family=tweedie(var.power=gamma,link.power=0),data=base),newdata=data.frame(x=1),type="response")-y[x==1] Verreur = Vectorize(erreur)(Vgamma) plot(Vgamma,Verreur,type="l",lwd=3,ylim=c(.001,.32),xlab="power",ylab="error")

So my point is that the distribution is usually not the most important point on GLMs, even if chapters of books on GLMs are distribution based… But as mentioned in an another post, if you consider a nonlinear transformation, like we have with GAMs, the story is more complicated…

# Actuariat de l’Assurance Non-Vie #9

Cette semaine, nous avons fini les modèles de tarification, avec une extention sur les modèles globaux (sans passer par une séparation entre fréquence et coût moyen), et sur le choix de variables, et le choix de modèles. Les slides introductifs sont en ligne.

# Actuariat de l’Assurance Non-Vie #9

Pour le neuvième chapitre du cours d’actuariat de l’assurance non-vie à l’ENSAE, un petit fourre-tout avant d’attaquer la modélisation du passif, en parlant un peu de modèles Tweedie (modèle collectif vs. modèles individuels), de choix de variables, et de choix de modèles. Les slides sont en ligne (la version pdf téléchargeable est comme souvent plus complète que celle sur slideshare)

# “Improving Segmentation” (using Lorenz curves, or sort of)

This afternoon, André did send me an interesting graph about the use of Lorenz curve in the context of insurance pricing (and modeling)

It is some sort of Lorenz curve, upside-down, with on the x-axis the proportion of the population, and on the y-axis the proportion of the losses. The important point is that the population is sorted according the their risk, i.e. their premium. The code to generate such a curve is actually quite simple,

L <- function(u,varx="premium",vary="losses"){
base=base[order(base[,varx],decreasing=TRUE),]
base$cum=(1:nrow(base))/nrow(base) return(sum(base[base$cum<=u,vary])/
sum(base[,vary]))}

vu=seq(0,1,by=.01)
vv=Vectorize(function(u) L(u))(vu)

My concern was more on two labels on the figure, with on the top-left “perfect pricing” and on the first diagonal “average pricing“. What could that possibly mean? Is there even such a thing as a “perfect pricing“? In order to understand what we plot here, let us generate some dataset, and fit some model. Including things that might be seen as the “perfect model“: the price base on the parameters used to generate the data, and the model used to generate the data, fitted on the data.