Studying joint effects in a regression
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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,
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