I am new to splines and I have to develop a model with interaction between splines of 2 continuous variables. I am not sure which of the 2 models are correct and what are the interpretation in each of the 2 models? I am not able to find any information online that can guide me to what is appropriate.
Model 1: Y= spline (x) + spline(z) + spline(x) * spline(z)
Model 2: Y= spline (x) + spline(z) + spline(x*z)
How are these 2 models different and what are their interpretations?
A simple multiplication is only one form of interaction. Forming splines of such oversimplified expressions often don’t fit the data and are dependent on the variables’ measurement origin or of how they are centered. Concentrate on tensor splines, i.e., take cross-products of all spline basis functions as in Model 1.
Here is more information including how to delete selected terms of the products of spline functions to put restrictions on the interaction surface and to reduce the number of parameters to estimates.
Probably a typo but you have y on both sides of the equation. That will not work well. Beyond that:
I don't think it's good to classify these as "correct" or "incorrect"; I'd use "useful" and "not useful". As for interpretation, I think graphics are the way to go. Make graphs of x1, x2, and predicted y for each model. Scatter plots, plots of the residuals, etc.
The numerical output from spline analysis is very opaque (at least to me, maybe some people are smart enough to get it).
