# The correct use of tensor product in gam (mgcv) function

I want to resurrect a question that I asked two months ago (Comparing gam models using ti( )), but adding more explanations.

• The aim of my analyses was to compare several gam models with different combinations of independent variables.

• My analyses are done in R using the gam function from the mgcv package

• Some of the models include interactions between some of the independent variables and in such a case, I use the following gam structure. gam(Y ~ ti(X1) + ti(X2) + ti(X1,X2), data = dat). The ti() function is a tensor product that is appropriate when interaction terms and main effects occur simultaneously (https://stat.ethz.ch/R-manual/R-devel/library/mgcv/html/te.html).

• Models that do not include interactions can be written as gam(Y ~ ti(X1) + ti(X2), data = dat) or gam(Y ~ s(X1) + s(X2), data = dat) with s() a function providing a smoother of the variable considered (https://stat.ethz.ch/R-manual/R-devel/library/mgcv/html/s.html)

I have two questions concerning this setup:

1. Is it correct using ti() (i.e. tensor product) when there is no interaction in the formula ?

(I am sure I will be blasted for this question, but if I use ti() instead of s() in those models their AIC value is better ... but see my second question)

1. Can I compare directly (e.g. using AIC for example) models fitted with ti() and models fitted with s() ?