The R Book

Generalized Additive Models

Generalized additive models (GAMs) are like GLMs in that they can have different error structures and different link functions to deal with count data or proportion data. What makes them different is that the shape of the relationship between y and a continuous variable x is not specified by some explicit functional form. Instead, non-parametric smoothers are used to describe the relationship. This is especially useful for relationships that exhibit complicated shapes, such as hump-shaped curves (see p. 666). The model looks just like a GLM, except that the relationships we want to be smoothed are prefixed by s: thus, if we had a three-variable multiple regression (with three continuous explanatory variables w, x and z) on count data and we wanted to smooth all three explanatory variables, we would write:

model<-gam(y~s(w)+s(x)+s(z),poisson)

These are hierarchical models, so the inclusion of a high-order interaction (such as A:B:C) necessarily implies the inclusion of all the lower-order terms marginal to it (i.e. A:B, A:C and B:C, along with main effects for A, B and C).

Because the models are nested, the more complicated model will necessarily explain at least as much of the variation as the simpler model (and usually more). What we want to know is whether the extra parameters in the more complex model are justified in the sense that they add significantly to the models explanatory power. If they do not, then parsimony requires that we accept the ...