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,