Consider the following additive model containing a smooth bivariate interaction term:
$y = f(x,z) + \epsilon$
where y is a continuous outcome variable, x and z are continuous predictor variables and $\epsilon$ is a random error term following a Normal distribution with mean 0 and unknown variance $\sigma^2$. Furthermore, $f$ is an unknown smooth bivariate function. In R, such a model would be fitted with the commands:
library(mgcv) model.1 <- gam(y ~ te(x,z), family=gaussian, data=data)
The mgcv help file https://www.rdocumentation.org/packages/mgcv/versions/1.8-24/topics/gam.models mentions that sometimes it is interesting to allow $f$ to have a main effects + interaction structure of the form $f(x,z) = f_1(x) + f_2(z) + f_3(x,z)$, meaning that the model would be expressed as:
$y = f_1(x) + f_2(z) + f_3(x,z) + \epsilon$
and fitted with the R command:
model.1 <- gam(y ~ ti(x) + ti(z) + ti(x,z), family=gaussian, data=data)
Q1: Generally speaking, under what circumstances would it be interesting to consider the main effects + interaction formulation for a bivariate smooth interaction? How would we know for a specific data set whether those circumstances were applicable? In particular, are those circumstances supposed to be driven by the data or by the research questions?
Q2: Assuming the circumstances in question are applicable for a data set, what is the interpretation of the main effects $f_1(x)$ and $f_2(z)$? For example, can we say that $f_1(x)$ represents the overall nonlinear effect of $x$ on the the outcome variable $y$, averaged across the values of $z$, while $f_2(x)$ represents the overall nonlinear effect of $z$ on the outcome variable $y$, averaged across the values of $x$? Also, would we need to worry in this context about the main smooth effects not necessarily being interpretable in the presence of a smooth interaction?
Q3: What is the interpretation of the smooth interaction term $f_3(x, z)$? Intuitively, how is this interpretation different from the interpretation of the smooth interaction term $f(x, z)$?
Many thanks for any insights you can share.