| gamm {mgcv} | R Documentation |
Fits the specified generalized additive mixed model (GAMM) to
data, by a call to lme in the normal errors identity link case, or by
a call to glmmPQL from the MASS library otherwise.
In the latter case estimates are only approximately MLEs. The routine is typically
much slower than gam, and not quite as numerically robust.
Smooths are specified as in a call to gam as part of the fixed
effects model formula, but the wiggly components of the smooth are treated as
random effects. The random effects structures and correlation structures
availabale for lme are used to specify other random effects and
correlations.
It is assumed that the random effects and correlation structures are employed primarily to model residual correlation in the data and that the prime interest is in inference about the terms in the fixed effects model formula including the smooths. For this reason the routine calculates a posterior covariance matrix for the coefficients of all the terms in the fixed effects formula, including the smooths.
gamm(formula,random=NULL,correlation=NULL,family=gaussian(), data=list(),weights=NULL,subset=NULL,na.action,knots=NULL, control=lmeControl(niterEM=3),niterPQL=20,verbosePQL=TRUE,...)
formula |
A GAM formula (see also gam.models).
This is exactly like the formula for a
glm except that smooth terms can be added to the right hand side of the
formula (and a formula of the form y ~ . is not allowed).
Smooth terms are specified by expressions of the form: s(var1,var2,...,k=12,fx=FALSE,bs="tp",by=a.var) where var1,
var2, etc. are the covariates which the smooth
is a function of and k is the dimension of the basis used to
represent the smooth term. If k is not
specified then k=10*3^(d-1) is used where d is the number
of covariates for this term. fx is used to indicate whether or
not this term has a fixed muber of degrees of freedom (fx=FALSE
to select d.f. by GCV/UBRE). bs indicates the basis to use, with
"cr" indicating cubic regression spline, "cc" a periodic
cubic regression spline, and "tp"
indicating thin plate regression spline: "cr" and "cc" can only be used
with 1-d smooths. Tensor product smooths are specified using te terms.
|
random |
The (optional) random effects structure as specified in a call to
lme: only the list form is allowed, to facilitate
manipulation of the random effects structure within gamm in order to deal
with smooth terms. |
correlation |
An optional corStruct object (see corClasses) as used to define correlation
structures in lme. |
family |
A family as used in a call to glm or gam.
The default gaussian with identity link causes gamm to fit by a direct call to
lme, otherwise glmmPQL from the MASS library is used. |
data |
A data frame containing the model response variable and
covariates required by the formula. By default the variables are taken
from environment(formula), typically the environment from
which gamm is called. |
weights |
prior weights on the data. Read the documentation for lme and glmmPQL very
carefully before even thinking about using this argument. |
subset |
an optional vector specifying a subset of observations to be used in the fitting process. |
na.action |
a function which indicates what should happen when the data contain `NA's. The default is set by the `na.action' setting of `options', and is `na.fail' if that is unset. The ``factory-fresh'' default is `na.omit'. |
knots |
this is an optional list containing user specified knot values to be used for basis construction.
For the cr basis the user simply supplies the knots to be used, and there must be the same number as the basis
dimension, k, for the smooth concerned. For the tp basis knots has two uses.
Firstly, for large datasets
the calculation of the tp basis can be time-consuming. The user can retain most of the advantages of the t.p.r.s.
approach by supplying a reduced set of covariate values from which to obtain the basis -
typically the number of covariate values used will be substantially
smaller than the number of data, and substantially larger than the basis dimension, k. The second possibility
is to avoid the eigen-decomposition used to find the t.p.r.s. basis altogether and simply use
the basis implied by the chosen knots: this will happen if the number of knots supplied matches the
basis dimension, k. For a given basis dimension the second option is
faster, but gives poorer results (and the user must be quite careful in choosing knot locations).
Different terms can use different
numbers of knots, unless they share a covariate.
|
control |
A list of fit control parameters for lme returned by
lmeControl. Note the default setting for the number of EM iterations
used by lme: high settings tend to lead to numerical problems because variance components
for smooth terms can legitimately be non-finite. |
niterPQL |
Maximum number of PQL iterations (if any). |
verbosePQL |
Should PQL report its progress as it goes along? |
... |
further arguments for passing on e.g. to lme |
The Bayesian model of spline smoothing introduced by Wahba (1983) and Silverman (1985) opens
up the possibility of estimating the degree of smoothness of terms in a generalized additive model
as variances of the wiggly components of the smooth terms treated as random effects. Several authors
have recognised this (see Wang 1998; Ruppert, Wand and Carroll, 2003) and in the normal errors, identity link case estimation can
be performed using general linear mixed effects modelling software such as lme. In the generalized case only
approximate inference is so far available, for example using the Penalized Quasi-Likelihood approach of Breslow
and Clayton (1993) as implemented in glmmPQL by Venables and Ripley (2002).
One advantage of this approach is that it allows correlated errors to be dealt with via random effects
or the correlation structures available in the nlme library.
Some brief details of how GAMs are represented as mixed models and estimated using lme or glmmPQL in gamm can be found in Wood (manuscript). In addition gamm obtains a posterior covariance matrix for the parameters of all the fixed effects and the smooth terms. The approach is similar to that described in (Lin & Zhang, 1999) - the covariance matrix of the data (or pseudodata in the generalized case) implied by the weights, correlation and random effects structure is obtained, based on the estimates of the parameters of these terms and this is used to obtain the posterior covariance matrix of the fixed and smooth effects.
The bases used to represent smooth terms are the same as those used in gam.
Returns a list with two items:
gam |
an object of class gam, less information relating to GCV/UBRE model selection. |
lme |
the fitted model object returned by lme or glmmPQL. Note that the model formulae and grouping
structures may appear to be rather bizarre, because of the manner in which the GAMM is split up and the calls to
lme and glmmPQL are constructed. |
lme and glmmPQL will not deal with offsets, so neither can gamm.
Models like s(z)+s(x)+s(x,z) are not currently supported.
gamm is not as numerically stable as gam: an lme call will occasionally fail. Experimenting with
niterEM in the control argument can sometimes help.
gamm is usually much slower than gam.
Note that the weights returned in the fitted GAM object are dummy, and not those used by the PQL iteration: this makes partial residual plots look odd.
Simon N. Wood simon@stats.gla.ac.uk
Breslow, N. E. and Clayton, D. G. (1993) Approximate inference in generalized linear mixed models. Journal of the American Statistical Association 88, 9-25.
Lin, X and Zhang, D. (1999) Inference in generalized additive mixed models by using smoothing splines. JRSSB. 55(2):381-400
Pinheiro J.C. and Bates, D.M. (2000) Mixed effects Models in S and S-PLUS. Springer
Ruppert, D., Wand, M.P. and Carroll, R.J. (2003) Semiparametric Regression. Cambridge
Silverman, B.W. (1985) Some aspects of the spline smoothing approach to nonparametric regression. JRSSB 47:1-52
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.
Wahba, G. (1983) Bayesian confidence intervals for the cross validated smoothing spline. JRSSB 45:133-150
Wood, S.N. (in press) Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of the American Statistical Association.
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114
Wood, S.N. (manuscript) Tensor product smooth interactions in Generalized Additive Mixed Models.
Wang, Y. (1998) Mixed effects smoothing spline analysis of variance. J.R. Statist. Soc. B 60, 159-174
http://www.stats.gla.ac.uk/~simon/
te, s, predict.gam,
plot.gam, summary.gam, gam.neg.bin,
vis.gam,pdTens,gamm.setup
library(mgcv)
set.seed(0)
n <- 400
sig <- 2
x0 <- runif(n, 0, 1)
x1 <- runif(n, 0, 1)
x2 <- runif(n, 0, 1)
x3 <- runif(n, 0, 1)
f <- 2 * sin(pi * x0)
f <- f + exp(2 * x1) - 3.75887
f <- f+0.2*x2^11*(10*(1-x2))^6+10*(10*x2)^3*(1-x2)^10-1.396
e <- rnorm(n, 0, sig)
y <- f + e
b <- gamm(y~s(x0)+s(x1)+s(x2)+s(x3))
plot(b$gam,pages=1)
b <- gamm(y~te(x0,x1)+s(x2)+s(x3))
op <- par(mfrow=c(2,2))
plot(b$gam)
par(op)
g<-exp(f/5)
y<-rpois(rep(1,n),g)
b2<-gamm(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson)
plot(b2$gam,pages=1)
# now an example with autocorrelated errors....
x <- 0:(n-1)/(n-1)
f <- 0.2*x^11*(10*(1-x))^6+10*(10*x)^3*(1-x)^10-1.396
e <- rnorm(n,0,sig)
for (i in 2:n) e[i] <- 0.6*e[i-1] + e[i]
y <- f + e
op <- par(mfrow=c(2,2))
b <- gamm(y~s(x,k=20),correlation=corAR1())
plot(b$gam);lines(x,f-mean(f),col=2)
b <- gamm(y~s(x,k=20))
plot(b$gam);lines(x,f-mean(f),col=2)
b <- gam(y~s(x,k=20))
plot(b);lines(x,f-mean(f),col=2)
par(op)
# and a "spatial" example
library(nlme);set.seed(1)
test1<-function(x,z,sx=0.3,sz=0.4)
{ (pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
}
n<-200
old.par<-par(mfrow=c(2,2))
x<-runif(n);z<-runif(n);
xs<-seq(0,1,length=30);zs<-seq(0,1,length=30)
pr<-data.frame(x=rep(xs,30),z=rep(zs,rep(30,30)))
truth <- matrix(test1(pr$x,pr$z),30,30)
contour(xs,zs,truth) # true function
f <- test1(x,z) # true expectation of response
cstr <- corGaus(.1,form = ~x+z)
cstr <- Initialize(cstr,data.frame(x=x,z=z))
V <- corMatrix(cstr) # correlation matrix for data
Cv <- chol(V)
e <- t(Cv) %*% rnorm(n)*0.05 # correlated errors
y <- f + e
b<- gamm(y~s(x,z,k=50),correlation=corGaus(.1,form=~x+z))
plot(b$gam) # gamm fit accounting for correlation
# overfits when correlation ignored.....
b1 <- gamm(y~s(x,z,k=50));plot(b1$gam)
b2 <- gam(y~s(x,z,k=50));plot(b2)
par(old.par)