We’ve seen in the previous post (here)  how important the *-cartesian
product to model joint effected in the regression. Consider the case of
two explanatory variates, one continuous (, the age of the driver) and one qualitative (, gasoline versus diesel).

• The additive model

Assume here that

Then, given  (the exposure, assumed to be constant) and 

Thus, there is a multplicative effect of the qualitative variate.
> reg=glm(nbre~bs(ageconducteur)+carburant+offset(exposition),
+     data=sinistres,family=”poisson”)
> ageD=data.frame(ageconducteur=seq(17,90),carburant=”D”,exposition=1)
> ageE=data.frame(ageconducteur=seq(17,90),carburant=”E”,exposition=1)
> yD=predict(reg,newdata=ageD,type=”response”)
> yE=predict(reg,newdata=ageE,type=”response”)
> lines(ageD$ageconducteur,yD,col=”blue”,lwd=2)
> lines(ageE$ageconducteur,yE,col=”red”,lwd=2)

On the graph below, we can see that the ratio

is constant (and independent of the age ).
> plot(ageD$ageconducteur,yD/yE)

• The nonadditive model

In
order to take into accound a more complex (non constant) interaction
between the two explanatory variates, consider the following product
model,
 > reg=glm(nbre~bs(ageconducteur)*carburant+offset(exposition),
+     data=sinistres,family=”poisson”)
> ageD=data.frame(ageconducteur=seq(17,90),carburant=”D”,exposition=1)
> ageE=data.frame(ageconducteur=seq(17,90),carburant=”E”,exposition=1)
> yD=predict(reg,newdata=ageD,type=”response”)
> yE=predict(reg,newdata=ageE,type=”response”)
> lines(ageD$ageconducteur,yD,col=”blue”,lwd=2)
> lines(ageE$ageconducteur,yE,col=”red”,lwd=2)

Here, the ratio

is not constant any longer,

• Mixing additive and nonadditive

It
is also possible to consider a model in between: we believe that there
is no interaction for young people (say), while there is for older
ones. Assume that the beak occurs at age 50,
> reg=glm(nbre~bs(ageconducteur*(ageconducteur<50))+
+     bs(ageconducteur*(ageconducteur>=50))*carburant+offset(exposition),
+     data=sinistres,family=”poisson”)
> ageD=data.frame(ageconducteur=seq(17,90,by=.1),carburant=”D”,exposition=1)
> ageE=data.frame(ageconducteur=seq(17,90,by=.1),carburant=”E”,exposition=1)
> yD=predict(reg,newdata=ageD,type=”response”)
> yE=predict(reg,newdata=ageE,type=”response”)
> lines(ageD$ageconducteur,yD,col=”blue”,lwd=2)
> lines(ageE$ageconducteur,yE,col=”red”,lwd=2)

Here, the ratio

is constant for young people, while it will change for older ones,