> > I am trying to understand the relationships between: > > > > y~s(x1)+s(x2)+s(x3)+s(x4) > > > > and > > > > y~s(x1,x2,x3,x4) > > > > Does the latter contain the former? what about the smoothers of all > > interaction terms? The first says that you want a model E(y) = f_1(x_1) + f_2(x_2) + f_3(x_3) + f_4(x_4) (1) where the f_j are smooth functions. The additive decomposition is quite a strong assumption, since it assumes that the effect of x_j is not dependent on x_k unless j=k. The second model is just E(y) = f(x_1,x_2,x_3,x4) (2) where f is a smooth function. This looks very general, but actually `s' terms assume isotropic smoothness, which is also quite a strong assumption.
Now if I simply state that f and the f_j are `smooth functions', and leave it at that, then (2) would of course contain (1), but to actually estimate the models I need to state, mathematically, what I mean by `smooth'. Once I've done that I've pretty much determined the function spaces in which f and the f_j will lie, and in general (2) will no longer strictly contain (1). mgcv's `s' terms use a thin plate spline measure of smoothness for multivariate smooths, and this means that (1) will not be strictly nested within (2), since e.g. a 4D thin plate spline can not generally represent exactly what the sum of 4 1D splines can represent. If you want to acheive exact nesting then using tensor product smooths with something like y~te(x1)+te(x2)+te(x3)+te(x4) (3) y~te(x1,x2,x3,x4) (4) will do the trick (because the function space for (4) is built up from the function spaces used in (3)). As to where all the 2 and 3 way interactions have gone in (4)... it's just like ANOVA - if you put in a 4 way interaction then the lower order interactions are not identifiable, unless you choose to add constraints to make them so. `mgcv' will allow you add main effects and interactions, and will handle the constraints automatically, but if this sort of functional ANOVA is a major component of what you want to do, then it is probably worth checking out the gss package and Chong Gu's book on smoothing spline ANOVA. best, Simon -- > Simon Wood, Mathematical Sciences, University of Bath, Bath, BA2 7AY UK > +44 1225 386603 www.maths.bath.ac.uk/~sw283 ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.