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 gammPQL
(a modification of glmmPQL
from the MASS
library) otherwise.
In the latter case estimates are only approximately MLEs. The routine is typically
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.
To use this function effectively it helps to be quite familiar with the use of
gam
and lme
.
gamm(formula,random=NULL,correlation=NULL,family=gaussian(), data=list(),weights=NULL,subset=NULL,na.action,knots=NULL, control=nlme::lmeControl(niterEM=0,optimMethod="L-BFGS-B"), niterPQL=20,verbosePQL=TRUE,method="ML",...)
formula |
A GAM formula (see also formula.gam and gam.models ).
This is like the formula for a glm except that smooth terms (s and te )
can be added to the right hand side of the
formula. Note that id s for smooths and fixed smoothing parameters are
not supported. |
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. See example below. |
correlation |
An optional corStruct object
(see corClasses ) as used to define correlation
structures in lme . Any grouping factors in the formula for
this object are assumed to be nested within any random effect grouping
factors, without the need to make this explicit in the formula (this is
slightly different to the behaviour of lme ). See examples below. |
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 procided there is no offset term, otherwise
gammPQL is used. |
data |
A data frame or list 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 |
In the generalized case, weights with the same meaning as
glm weights. An lme type weights argument may only be
used in the identity link gaussian case, with no offset (see documentation for lme
for details of how to use such an 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. 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 : smooths are set up using custom pdMat classes,
which are currently not supported by the EM iteration code. Only increase this
number if you want to perturb the starting values used in model fitting
(usually to worse values!). The optimMethod option is only used if your
version of R does not have the nlminb optimizer function. |
niterPQL |
Maximum number of PQL iterations (if any). |
verbosePQL |
Should PQL report its progress as it goes along? |
method |
Which of "ML" or "REML" to use in the Gaussian
additive mixed model case when lme is called directly. Ignored in the
generalized case (or if the model has an offset), in which case gammPQL is used. |
... |
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 details of how GAMs are represented as mixed models and estimated using
lme
or gammPQL
in gamm
can be found in Wood (2004 ,2006a,b). 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
, although adaptive
smoothing bases are not available.
In the event of lme
convergence failures, consider
modifying option(mgcv.vc.logrange)
: reducing it helps to remove
indefiniteness in the likelihood, if that is the problem, but too large a
reduction can force over or undersmoothing. See notExp2
for more
information on this option. Failing that, you can try increasing the
niterEM
option in control
: this will perturb the starting values
used in fitting, but usually to values with lower likelihood! Note that this
version of gamm
works best with R 2.2.0 or above and nlme
, 3.1-62 and above,
since these use an improved optimizer.
Returns a list with two items:
gam |
an object of class gam , less information relating to
GCV/UBRE model selection. At present this contains enough information to use
predict , summary and print methods and vis.gam ,
but not to use e.g. the anova method function to compare models. |
lme |
the fitted model object returned by lme or gammPQL . 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 gammPQL are constructed. |
gamm
assumes that you know what you are doing! For example, unlike
glmmPQL
from MASS
it will return the complete lme
object
from the working model at convergence of the PQL iteration, including the `log
likelihood', even though this is not the likelihood of the fitted GAMM.
The routine will be very slow and memory intensive if correlation structures are used for the very large groups of data. e.g. attempting to run the spatial example in the examples section with many 1000's of data is definitely not recommended: often the correlations should only apply within clusters that can be defined by a grouping factor, and provided these clusters do not get too huge then fitting is usually possible.
Models must contain at least one random effect: either a smooth with non-zero
smoothing parameter, or a random effect specified in argument random
.
gamm
is not as numerically stable as gam
: an lme
call
will occasionally fail. See details section for suggestions.
gamm
is usually much slower than gam
, and on some platforms you may need to
increase the memory available to R in order to use it with large data sets
(see mem.limits
).
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.
Note that the gam
object part of the returned object is not complete in
the sense of having all the elements defined in gamObject
and
does not inherit from glm
: hence e.g. multi-model anova
calls will not work.
The parameterization used for the smoothing parameters in gamm
, bounds
them above and below by an effective infinity and effective zero. See
notExp2
for details of how to change this.
Linked smoothing parameters and adaptive smoothing are not supported.
Simon N. Wood simon.wood@r-project.org
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. (2004) Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of the American Statistical Association. 99:673-686
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114
Wood, S.N. (2006a) Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4):1025-1036
Wood S.N. (2006b) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press.
Wang, Y. (1998) Mixed effects smoothing spline analysis of variance. J.R. Statist. Soc. B 60, 159-174
http://www.maths.bath.ac.uk/~sw283/
magic
for an alternative for correlated data,
te
, s
,
predict.gam
,
plot.gam
, summary.gam
, negbin
,
vis.gam
,pdTens
library(mgcv) ## simple examples using gamm as alternative to gam set.seed(0) dat <- gamSim(1,n=400,scale=2) b <- gamm(y~s(x0)+s(x1)+s(x2)+s(x3),data=dat) plot(b$gam,pages=1) summary(b$lme) # details of underlying lme fit summary(b$gam) # gam style summary of fitted model anova(b$gam) b <- gamm(y~te(x0,x1)+s(x2)+s(x3),data=dat) op <- par(mfrow=c(2,2)) plot(b$gam) par(op) rm(dat) ## Add a factor to the linear predictor, to be modelled as random dat <- gamSim(6,n=400,scale=.2,dist="poisson") b2<-gamm(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson, data=dat,random=list(fac=~1)) plot(b2$gam,pages=1) fac <- dat$fac rm(dat) ## now an example with autocorrelated errors.... n <- 400;sig <- 2 x <- 0:(n-1)/(n-1) f <- 0.2*x^11*(10*(1-x))^6+10*(10*x)^3*(1-x)^10 e <- rnorm(n,0,sig) for (i in 2:n) e[i] <- 0.6*e[i-1] + e[i] y <- f + e op <- par(mfrow=c(2,2)) b <- gamm(y~s(x,k=20),correlation=corAR1()) plot(b$gam);lines(x,f-mean(f),col=2) b <- gamm(y~s(x,k=20)) plot(b$gam);lines(x,f-mean(f),col=2) b <- gam(y~s(x,k=20)) plot(b);lines(x,f-mean(f),col=2) ## more complicated autocorrelation example - AR errors ## only within groups defined by `fac' e <- rnorm(n,0,sig) for (i in 2:n) e[i] <- 0.6*e[i-1]*(fac[i-1]==fac[i]) + e[i] y <- f + e b <- gamm(y~s(x,k=20),correlation=corAR1(form=~1|fac)) plot(b$gam);lines(x,f-mean(f),col=2) par(op) ## more complex situation with nested random effects and within ## group correlation set.seed(0) n.g <- 10 n<-n.g*10*4 ## simulate smooth part... dat <- gamSim(1,n=n,scale=2) f <- dat$f ## simulate nested random effects.... fa <- as.factor(rep(1:10,rep(4*n.g,10))) ra <- rep(rnorm(10),rep(4*n.g,10)) fb <- as.factor(rep(rep(1:4,rep(n.g,4)),10)) rb <- rep(rnorm(4),rep(n.g,4)) for (i in 1:9) rb <- c(rb,rep(rnorm(4),rep(n.g,4))) ## simulate auto-correlated errors within groups e<-array(0,0) for (i in 1:40) { eg <- rnorm(n.g, 0, sig) for (j in 2:n.g) eg[j] <- eg[j-1]*0.6+ eg[j] e<-c(e,eg) } dat$y <- f + ra + rb + e dat$fa <- fa;dat$fb <- fb ## fit model .... b <- gamm(y~s(x0,bs="cr")+s(x1,bs="cr")+s(x2,bs="cr")+ s(x3,bs="cr"),data=dat,random=list(fa=~1,fb=~1), correlation=corAR1()) plot(b$gam,pages=1) ## and a "spatial" example... library(nlme);set.seed(1);n <- 200 dat <- gamSim(2,n=n,scale=0) ## standard example attach(dat) old.par<-par(mfrow=c(2,2)) contour(truth$x,truth$z,truth$f) ## true function f <- data$f ## true expected response ## Now simulate correlated errors... cstr <- corGaus(.1,form = ~x+z) cstr <- Initialize(cstr,data.frame(x=data$x,z=data$z)) V <- corMatrix(cstr) ## correlation matrix for data Cv <- chol(V) e <- t(Cv) %*% rnorm(n)*0.05 # correlated errors ## next add correlated simulated errors to expected values data$y <- f + e ## ... to produce response b<- gamm(y~s(x,z,k=50),correlation=corGaus(.1,form=~x+z), data=data) plot(b$gam) # gamm fit accounting for correlation # overfits when correlation ignored..... b1 <- gamm(y~s(x,z,k=50),data=data);plot(b1$gam) b2 <- gam(y~s(x,z,k=50),data=data);plot(b2) par(old.par)