Specifying generalized additive models

Description

This page is intended to provide some more information on how to specify GAMs. Assume that in general we are interested in modelling some response variable y with an exponential family distribution (e.g. Normal, Poisson, binomial, etc.) using predictor variables x1,x2,..., and let m=E(y). A typical GAM might be:

g(m)=b0 + b1x1 + b2x2 + f1(x3) + f2(x4,x5)

where g is a smooth monotonic `link' function, and f1 and f2 are smooth functions. The key idea here is that the dependence of the response on the predictors can be represented as a parametric sub-model plus the sum of some smooth fucntions of one or more of the predictor variables. Thus the model is quite flexible relative to strictly parametric linear or generalized linear models, but still has much more structure than the completely general model that says that the response is just some smooth function of all the covariates.

Note one important point. In order for the model to be identifiable the smooth functions have to be constrained to have zero mean (usually taken over the set of covariate values). Such constraints are always applied by gam.

Specification of the distribution and link function is done using the family argument to gam and works in the same way as for glm. This page hence assumes the default identity link normal error family, since the generalizations are easy.

Starting with the model given above then, the gam formula would be
y~x1+x2+s(x3)+s(x4,x5).
This would use the default basis for the smooths (a thin plate regression spline basis for each), with automatic selection of the effective degrees of freedom for both smooths. The dimension of the smoothing basis is given a default value as well (the dimension of the basis sets an upper limit on the maximum possible degrees of freedom for the basis - the limit is typically one less than basis dimension). Full details of how to control smooths are given in s and te. For the moment suppose that we would like to change the basis of the first smooth to a cubic regression spline basis with a dimension of 20, while fixing the second term at 25 degrees of freedom. The appropriate formula would be:
y~x1+x2+s(x3,bs="cr",k=20)+s(x4,x5,k=26,fx=T).

Now consider some more unusual models. Consider a model in which y is a smooth function of x except at a point x* where the function jumps discontinuously. This model can be written as:

E(y) = b0 + b1 h(x*,x) + f1(x)

where h is a step function jumping from 0 to 1 at x*. The way to fit this model is to create a variable h which is zero for all x less than x* and one otherwise. Then the model formula is:
y~h+s(x).

Another situation that occurs quite often is the one in which we would like to find out if the model:

E(y)=f(x,z)

is really necessary or whether:

E(y)=f1(x)+f2(z)

wouldn't do just as well. One way to do this is to look at the results of fitting:
y~s(x)+s(z)+s(x,z).
gam automatically generates side conditions to make this model identifiable. You can also estimate `overlapping' models like:
y~s(x,z)+s(z,v).

Sometimes models of the form:

E(y)=b0+f(x)z

need to be estimated (where f is a smooth function, as usual.) The appropriate formula is:
y~z+s(x,by=z)
- the by argument ensures that the smooth function gets multiplied by covariate z, but GAM smooths are centred (average value zero), so the z+ term is needed as well (f is being represented by a constant plus a centred smooth). If we'd wanted:

E(y)=f(x)z

then the appropriate formula would be:
y~z+s(x,by=z)-1.

The by mechanism also allows models to be estimated in which the form of a smooth depends on the level of a factor, but to do this the user must generate the dummy variables for each level of the factor. Suppose for example that fac is a factor with 3 levels 1, 2, 3, and at each level of this factor ther response depends smoothly on a variable x in a manner that is level dependent. Three dummy variables fac.1, fac.2, fac.3, can be generated for the factor (e.g. fac.1<-as.numeric(fac==1)). Then the model formula would be:
y~fac+s(x,by=fac.1)+s(x,by=fac.2)+s(x,by=fac.3).

In the above examples the smooths of more than one covariate have all employed single penalty thin plate regression splines. These isotropic smooths are not alway appropriate: if variables are not naturally `well scaled' relative to each other then it is often preferable to use tensor product smooths, with a wiggliness penalty for each covariate of the term. See te for examples.

WARNING

There are no identifiability checks made between the smooth and parametric parts of a gam formula, and any such lack of identifiability will cause problems if the underlying fitting routine is mgcv (not the default).

Author(s)

Simon N. Wood simon.wood@r-project.org

Examples

set.seed(10)
n<-400
sig2<-4
x0 <- runif(n, 0, 1)
x1 <- runif(n, 0, 1)
x2 <- runif(n, 0, 1)
pi <- asin(1) * 2
f1 <- 2 * sin(pi * x2)
f2 <-  exp(2 * x2) - 3.75887
f3 <-  0.2 * x2^11 * (10 * (1 - x2))^6 + 
       10 * (10 * x2)^3 * (1 - x2)^10 - 1.396
e <- rnorm(n, 0, sqrt(abs(sig2)))
# A continuous `by' variable example.... 
y <- f3*x1 + e
b<-gam(y~x1-1+s(x2,by=x1))
plot(b,pages=1)
summary(b)
# A dummy `by' variable example (with a spurious covariate x0)
fac<-as.factor(c(rep(1,100),rep(2,100),rep(3,200)))
fac.1<-as.numeric(fac==1);fac.2<-as.numeric(fac==2);
fac.3<-as.numeric(fac==3)
y<-f1*fac.1+f2*fac.2+f3*fac.3+ e
b<-gam(y~fac+s(x2,by=fac.1)+s(x2,by=fac.2)+s(x2,by=fac.3)+s(x0))
plot(b,pages=1)
summary(b)