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=0,optimMethod="L-BFGS-B"),niterPQL=20,
verbosePQL=TRUE,method="ML",...)

Arguments

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.

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.

Value

Returns a list with two items:

WARNINGS

This version of gamm works best with nlme 3.1-62 and above and R 2.2.0 and above. This is because it is designed to work with the optimizers used from these versions onwards, rather than the earlier default optimizers.

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. 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.

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. 99:673-686

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)