How to remove tensor interaction in gam function?

[Nataliannash]

Dear Gavin,

I have a question about removing the tensor interaction in gam function, and I would appreciate it if you could help me.
In example:
fit <- gam(Y ~ s(X1) + s(X2) + s(X3) +s(X4) + ti(X1,X2,X3) , data=data), I used ti() function because there is interaction between the three variables (X1,X2,X3).

How should I change the gam function if I want that gam just consider the pure effects of these three variables without their interaction?

Some references said that if there is interaction between variables, removing ti() may cause repeating same covariates multiple times.

I tried fit <- gam(Y ~ s(X1) + s(X2) + s(X3) +s(X4) + ti(X1,X2,X3 , np=FALSE) , data=data), and compared it by "np=TRUE" but nothing changed in results.

Kind regards,
Nat

[gavinsimpson]

Instead of opening an Issue (which is for bug reports or feature requests) please use the Discussions tab.

Technically, your model should be

m1 <- gam(Y ~ s(X1) + s(X2) + s(X3) + s(X4) +      # first order terms
            ti(X1, X2) + ti(X1, X3) + ti(X2, X3) + # second order terms
            ti(X1, X2, X3),                        # third order term
          data=data, family = FOO, method = "REML")

Now you can consider dropping the ti(X1, X2, X3), and from there consider if you need any of the second order interactions.

Adding select = TRUE to the model if you want to do model selection.

If your question is, can you fit

m2 <- gam(Y ~ s(X1) + s(X2) + s(X3) + s(X4),
          data=data, family = FOO, method = "REML")

and compare it with the above, yes, you can.

However, if you believe there may be interactions between your covariates and you have sufficient data to estimate those higher order terms, and if you are interested in anything other than prediction (i.e. you want to estimate effects or do inference), I would just fit the full model (with or without select = TRUE) and just work with that model (m1).

Either the sources you mention are confused or you have misunderstood. It would be a problem to fit a simpler version of your model (without the third order term) using te():

m3 <- gam(Y ~ s(X4) +
            te(X1, X2) + te(X1, X3) + te(X2, X3), # second order terms
          data=data, family = FOO, method = "REML")

because in m3 we have full tensor products, including main effects, of the stated variables, which means the main effect are included in the model twice for each of X1, X2, and X3. The ti() smooths are a way to avoid that, such that we would fit:

m4 <- gam(Y ~ s(X1) + s(X2) + s(X3) + s(X4) +      # first order terms
            ti(X1, X2) + ti(X1, X3) + ti(X2, X3),  # second order terms
          data=data, family = FOO, method = "REML")

and except for some additional smoothing parameters we would get the model implied by m3 but in a more computationally-stable way.