gamm {mgcv}R Documentation

Generalized Additive Mixed Models

Description

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 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.

Usage

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,...)

Arguments

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: see s for full details, but note that the default "tp" can be slow with large data sets. 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. 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 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. See documentation for lme for details of how to use 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

Details

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 (2004a,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.

Value

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 comapre models.
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.

WARNINGS

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.

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, 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.

Author(s)

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

References

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. (2004a) Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of the American Statistical Association. 99:637-686

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

Wood, S.N. (2004b) Low rank scale invariant tensor product smooths for Generalized Additive Mixed Models. Technical Report of the Department of Statistics, University of Glasgow, UK.

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/

See Also

te, s, predict.gam, plot.gam, summary.gam, gam.neg.bin, vis.gam,pdTens,gamm.setup

Examples

library(mgcv)
## simple examples using gamm as alternative to gam
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$gam,pages=1)
summary(b$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)) 
op <- par(mfrow=c(2,2))
plot(b$gam)
par(op)

## Add a factor to the linear predictor, to be modelled as random
fac <- rep(1:4,n/4)
f <- f + fac*3
fac<-as.factor(fac)

g<-exp(f/5)
y<-rpois(rep(1,n),g)
b2<-gamm(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson,random=list(fac=~1))
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)

## 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
sig <- 2
## simulate smooth part
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
## 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)
}
y <- f + ra + rb + e
dat<-data.frame(y=y,x0=x0,x1=x1,x2=x2,x3=x3,fa=fa,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)
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
## Now simulate correlated errors...
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
## next add correlated simulated errors to expected values
y <- f + e ## ... to produce response
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)


[Package mgcv version 1.2-3 Index]