\Header{gam.models}{Specifying generalized Additive Models.}
\keyword{models}{gam.models}
\keyword{regression}{gam.models}
\begin{Description}\relax
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 \eqn{y_i}{y} with an exponential
family distribution (e.g. Normal, Poisson, binomial, etc.) using predictor variables
\eqn{x_{1i}, x_{2i}, \ldots}{x1,x2,...}, and let \eqn{\mu_i \equiv
E(y_i)}{m=E(y)}. A typical GAM might be:
\deqn{g(\mu_i) = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} +
f_1(x_{3i})+f_2(x_{4i},x_{4i})}{g(m)=b0 + b1x1 + b2x2 + f1(x3) +
f2(x4,x5) }
where \eqn{g}{} is a smooth monotonic "link" function, and \eqn{f_1}{f1}
and \eqn{f_2}{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 \code{gam}.

Specification of the distribution and link function is done using the \code{\Link{family}}
argument to \code{\Link{gam}} and works in the same way as for
\code{\Link{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 \code{gam} formula would
be \\
\code{y\textasciitilde{}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
\code{\Link{s}}. 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:\\
\code{y\textasciitilde{}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
\eqn{y}{y} is a smooth function of \eqn{x}{x} except at a point
\eqn{x^*}{x*} where the function jumps discontinuously. This model can
be written as:
\deqn{E(y_i)= \beta_0+\beta_1h(x*,x_i)+f_1(x_i)}{E(y) = b0 + b1
h(x*,x) + f1(x)}
where \eqn{h}{} is a step function jumping from 0 to 1 at
\eqn{x^*}{x*}. The way to fit this model is to create a variable
\code{h} which is zero for all \code{x} less than \eqn{x^*}{x*} and
one otherwise. Then the model formula is:\\
\code{y\textasciitilde{}h+s(x)}. \\

Another situation that occurs quite often is the one in which we
would like to find out if the model:
\deqn{E(y_i) = f(x_i,z_i)}{E(y)=f(x,z)}
is really necessary or whether:
\deqn{E(y_i)=f_1(x_i)+f_2(z_i)}{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:\\
\code{y\textasciitilde{}s(x)+s(z)+s(x,z)}.\\
\code{gam} automatically generates side conditions to make this model
identifiable. You can also estimate "overlapping" models like:\\
\code{y\textasciitilde{}s(x,z)+s(z,v)}.

Sometimes models of the form:
\deqn{E(y_i) = \beta_0+ f(x_i)z_i}{E(y)=b0+f(x)z}
need to be estimated (where \eqn{f}{} is a smooth function, as usual.)
The appropriate formula is:\\
\code{y\textasciitilde{}z+s(x,by=z)}\\
- the \code{by} argument ensures that the smooth function gets multiplied by
covariate \code{z}, but GAM smooths are centred (average value zero),
so the \code{z+} term is needed as well (\eqn{f}{} is being
represented by a constant plus a centred smooth). If we'd wanted:
\deqn{E(y_i) = f(x_i)z_i}{E(y)=f(x)z}
then the appropriate formula would be:\\
\code{y\textasciitilde{}z+s(x,by=z)-1}.\\

The \code{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 \code{fac} is a factor with 3 levels
\code{1}, \code{2}, \code{3},
and at each level of this factor ther response depends smoothly on a
variable \code{x} in a manner that is level dependent. Three dummy
variables \code{fac.1}, \code{fac.2}, \code{fac.3}, can be generated for the factor   
(e.g. \code{fac.1<-as.numeric(fac==1)}). Then the model formula would
be:\\
\code{y\textasciitilde{}fac+s(x,by=fac.1)+s(x,by=fac.2)+s(x,by=fac.3)}.\\\end{Description}
\begin{Section}{WARNING}
There are no identifiability checks made between the smooth and
parametric parts of a gam formula.\end{Section}
\begin{Author}\relax
Simon N. Wood \email{simon@stats.gla.ac.uk}\end{Author}
\begin{Examples}
\begin{ExampleCode}
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)
\end{ExampleCode}
\end{Examples}

