gam() attempts to find the appropriate smoothness for each applicable model
term using Generalized Cross Validation (GCV) or an Un-Biased Risk Estimator
(UBRE). The latter is used in cases in
which the scale parameter is assumed known, in which case it is very similar
to AIC or Mallows' Cp. GCV and UBRE are covered in Craven and Wahba (1979) and Wahba (1990), see
gam.method for more detail about the numerical optimization
approaches available.
Automatic smoothness selection is unlikely to be successful with few data, particularly with
multiple terms to be selected. In addition GCV and UBRE/AIC score can occasionally
display local minima that can trap the minimisation algorithms. GCV/UBRE/AIC
scores become constant with changing smoothing parameters at very low or very
high smoothng parameters, and on occasion these `flat' regions can be
separated from regions of lower score by a small `lip'. This seems to be the
most common form of local minimum, but is usually avoidable by avoiding
extreme smoothing parameters as starting values in optimization, and by
avoiding big jumps in smoothing parameters while optimizing. Never the less, if you are
suspicious of smoothing parameter estimates, try changing fit method (see
gam.method) and see if the estimates change, or try changing
some or all of the smoothing parameters `manually' (argument sp of gam,
or sp arguments to s or te).
In general the most logically consistent method to use for deciding which
terms to include in the model is to compare GCV/UBRE scores for models with
and without the term. When UBRE is the smoothness selection method this will
give the same result as comparing by AIC (the AIC in this case
uses the model EDF in place of the usual model DF). Similarly, comparison via
GCV score and via AIC seldom yields different answers. Note that the negative
binomial with estimated theta parameter is a special case: the GCV
score is not informative, because of the theta estimation
scheme used.
More generally the score for the model with a smooth
term can be compared to the score for the model with
the smooth term replaced by appropriate parametric terms. Candidates for
removal can be identified by reference to the approximate p-values provided
by summary.gam. Candidates for replacement by parametric terms are
smooth terms with estimated degrees of freedom close to their
minimum possible.
One appealing approach to model selection is via shrinkage. Smooth classes
cs.smooth and tprs.smooth (specified by "cs" and
"ts" respectively) have smoothness penalties which include a small
shrinkage component, so that for large enough smoothing parameters the smooth
becomes identically zero. This allows automatic smoothing parameter selection
methods to effectively remove the term from the model altogether. The
shrinkage component of the penalty is set at a level that usually makes negligable
contribution to the penalization of the model, only becoming effective when
the term is effectively `completely smooth' according to the conventional penalty.
Note that GCV and UBRE are not appropriate for comparing models using different families: in that case AIC should be used.
Venables and Ripley (1999) Modern Applied Statistics with S-PLUS
Wahba (1990) Spline Models of Observational Data. SIAM.
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114
Wood, S.N. (2008) Fast stable direct fitting and smoothness selection for generalized additive models. J.R.Statist. Soc. B 70(3):495-518
## an example of GCV based model selection
library(mgcv)
set.seed(0);n<-400
dat <- gamSim(1,n=n,scale=2) ## simulate data
dat$x4 <- runif(n, 0, 1);dat$x5 <- runif(n, 0, 1) ## spurious
## Note the increased gamma parameter below to favour
## slightly smoother models (See Kim & Gu, 2004, JRSSB)...
b<-gam(y~s(x0,bs="ts")+s(x1,bs="ts")+s(x2,bs="ts")+
s(x3,bs="ts")+s(x4,bs="ts")+s(x5,bs="ts"),gamma=1.4,data=dat)
summary(b)
plot(b,pages=1)Run the code above in your browser using DataLab