# I Fought the (distribution) Law (and the Law did not win)

A few days ago, I was asked if we should spend a lot of time to choose the distribution we use, in GLMs, for (actuarial) ratemaking. On that topic, I usually claim that the family is not the most important parameter in the regression model. Consider the following dataset

> db <- data.frame(x=c(1,2,3,4,5),y=c(1,2,4,2,6))
> plot(db,xlim=c(0,6),ylim=c(-1,8),pch=19)

To visualize a regression model, use the following code

> nd=data.frame(x=seq(0,6,by=.1))
> add_predict = function(reg){
+ prd1=predict(reg,newdata=nd,se.fit = TRUE,type="response")
+ y1=prd1$fit + y1_upp=prd1$fit+prd1$residual.scale*1.96* prd1$se.fit
+ y1_low=prd1$fit-prd1$residual.scale*1.96*
prd1$se.fit + polygon(c(nd$x,rev(nd$x)),c(y1_upp, rev(y1_low)),col="light green",angle=90, density=40,border=NA) + lines(nd$x,y1,col="red",lwd=2)
+ }

For instance, with a Poisson regression (with a log link function) we get

> plot(db)
> reg1=glm(y~x,family=poisson(link="log"),
+ data=db)
> add_predict(reg1)


while, with a Gaussian regresion (but still with a log link function), we get

> plot(db)
> reg2=glm(y~x,family=gaussian(link="log"),
+ data=db)
> add_predict(reg2)

If we just care about the expected value of our prediction, the output is more or less the same

> plot(db)
> lines(nd$x,predict(reg1,newdata=nd, + type="response"),col="red",lwd=1.5) > lines(nd$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)

So, indeed, forget about the (distribution) law when running a GLM. Not convinced? Consider – on the same dataset – a Poisson regression (with an identity link function this time)

> plot(db)
> reg1=glm(y~x,family=poisson(link="identity"),
+ data=db)
> add_predict(reg1)


while, with a Gaussian regresion (but still with an identity link function), we get

> plot(db)
> reg2=glm(y~x,family=gaussian(link="identity"),
+ data=db)
> add_predict(reg2)

Again, if we just plot the expected value of our prediction, the output is more or less the same

> plot(db)
> lines(nd$x,predict(reg1,newdata=nd, + type="response"),col="red",lwd=1.5) > lines(nd$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)

So clearly, the simplistic message you should not care too much about the (distribution) law seems to be valid…

# GLM, non-linearity and heteroscedasticity

Last week in the non-life insurance course, we’ve seen the theory of the Generalized Linear Models, emphasizing the two important components

• the link function (which is actually the key component in predictive modeling)
• the distribution, or the variance function

Just to illustrate, consider my favorite dataset

­lin.mod = lm(dist~speed,data=cars)

A linear model means here

$Y_i=\beta_0+\beta_1 X_i +\varepsilon_i$

where the residuals are assumed to be centered, independent, and with identical variance. If we visualize that linear regression, we usually see something like that

The idea here (in GLMs) is to assume

$Y\vert X=x\sim\mathcal{N}( \beta_0+\beta_1 x,\sigma^2)$

which will produce the same model as the one describe previously, based on some error term. That model can be visualized below,

attach(cars)
n=2
X= cars$speed Y=cars$dist
df=data.frame(X,Y)
vX=seq(min(X)-2,max(X)+2,length=n)
vY=seq(min(Y)-15,max(Y)+15,length=n)
mat=persp(vX,vY,matrix(0,n,n),zlim=c(0,.1),theta=-30,ticktype ="detailed", box = FALSE)
reggig=glm(Y~X,data=df,family=gaussian(link="identity"))
x=seq(min(X),max(X),length=501)
C=trans3d(x,predict(reggig,newdata=data.frame(X=x),type="response"),rep(0,length(x)),mat)
lines(C,lwd=2)
sdgig=sqrt(summary(reggig)$dispersion) x=seq(min(X),max(X),length=501) y1=qnorm(.95,predict(reggig,newdata=data.frame(X=x),type="response"), sdgig) C=trans3d(x,y1,rep(0,length(x)),mat) lines(C,lty=2) y2=qnorm(.05,predict(reggig,newdata=data.frame(X=x),type="response"), sdgig) C=trans3d(x,y2,rep(0,length(x)),mat) lines(C,lty=2) C=trans3d(c(x,rev(x)),c(y1,rev(y2)),rep(0,2*length(x)),mat) polygon(C,border=NA,col="yellow") C=trans3d(X,Y,rep(0,length(X)),mat) points(C,pch=19,col="red") n=8 vX=seq(min(X),max(X),length=n) mgig=predict(reggig,newdata=data.frame(X=vX)) sdgig=sqrt(summary(reggig)$dispersion)
for(j in n:1){
stp=251
x=rep(vX[j],stp)
y=seq(min(min(Y)-15,qnorm(.05,predict(reggig,newdata=data.frame(X=vX[j]),type="response"), sdgig)),max(Y)+15,length=stp)
z0=rep(0,stp)
z=dnorm(y, mgig[j], sdgig)
C=trans3d(c(x,x),c(y,rev(y)),c(z,z0),mat)
polygon(C,border=NA,col="light blue",density=40)
C=trans3d(x,y,z0,mat)
lines(C,lty=2)
C=trans3d(x,y,z,mat)
lines(C,col="blue")}

We do have two parts here: the linear increase of the average, $\mathbb{E}(Y\vert X=x)=\beta_0+\beta_1 x$ and the constant variance of the normal distribution $\text{Var}(Y\vert X=x)=\sigma^2$.

On the other hand, if we assume a Poisson regression,

poisson.reg = glm(dist~speed,data=cars,family=poisson(link="log"))

we have something like

This time, two things have changed simultaneously: our model is no longer linear, it is an exponential one $\mathbb{E}(Y\vert X=x)=e^{\beta_0+\beta_1 x}$, and the variance is also increasing with the explanatory variable $\text{Var}(Y\vert X=x)=e^{\beta_0+\beta_1 x}$, since with a Poisson regression,

$Y\vert X=x\sim\mathcal{P}(e^{ \beta_0+\beta_1 x})$

If we adapt the previous code, we get

The problem is that we changed two things when we introduced the Poisson regression from the linear model. So let us look at what happens when we change the two components independently. First, we can change the link function, with a Gaussian model but this time a multiplicative model (with a logarithm link function)

gaussian.reg = glm(dist~speed,data=cars,family=gaussian(link="log"))

which is still, here, an homoscedasctic model, but this time non-linear. Or we can change the link function in the Poisson regression, to get a linear model, but heteroscedastic

poisson.lin = glm(dist~speed,data=cars,family=poisson(link="identity"))

So this is basically what GLMs are about….