| gamm4 {gamm4} | R Documentation |
Generalized Additive Mixed Models using lme4 and mgcv
Description
Fits the specified generalized additive mixed model (GAMM) to
data, by making use of the modular fitting functions provided
by lme4 (new version). For earlier lme4 versions modelling fitting is via
a call to lmer in the normal errors identity link case, or by
a call to glmer otherwise (see lmer). Smoothness selection is by REML in the Gaussian
additive case and (Laplace approximate) ML otherwise.
gamm4 is based on gamm from package mgcv, but uses lme4 rather than
nlme as the underlying fitting engine via a trick due to Fabian Scheipl.
gamm4 is more robust numerically than gamm, and by avoiding PQL gives better
performance for binary and low mean count data. Its main disadvantage is that it can not handle most multi-penalty
smooths (i.e. not te type tensor products or adaptive smooths) and there is
no facilty for nlme style correlation structures. Tensor product smoothing is available via
t2 terms (Wood, Scheipl and Faraway, 2013).
For fitting generalized additive models without random effects, gamm4 is much slower
than gam and has slightly worse MSE performance than gam
with REML smoothness selection. For fitting GAMMs with modest numbers of i.i.d. random coefficients
then gamm4 is slower than gam (or bam for large data
sets). gamm4 is most useful when the random effects are not i.i.d., or when there are large
numbers of random coeffecients (more than several hundred), each applying to only a small proportion
of the response data.
To use this function effectively it helps to be quite familiar with the use of
gam and lmer.
Usage
gamm4(formula,random=NULL,family=gaussian(),data=list(),weights=NULL,
subset=NULL,na.action,knots=NULL,drop.unused.levels=TRUE,
REML=TRUE,control=NULL,start=NULL,verbose=0L,...)
Arguments
formula |
A GAM formula (see also |
random |
An optional formula specifying the random effects structure in |
family |
|
data |
A data frame or list containing the model response variable and
covariates required by the formula. By default the variables are taken
from |
weights |
a vector of prior weights on the observations. |
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. |
drop.unused.levels |
by default unused levels are dropped from factors before fitting. For some smooths involving factor variables you might want to turn this off. Only do so if you know what you are doing. |
REML |
passed on to |
control |
|
start |
|
verbose |
passed on to fitting |
... |
further arguments for passing on to model setup routines. |
Details
A generalized additive mixed model is a generalized linear mixed model in which the linear predictor
depends linearly on unknown smooth functions of some of the covariates (‘smooths’ for short). gamm4 follows the approach taken
by package mgcv and represents the smooths using penalized regression spline type smoothers, of
moderate rank. For estimation purposes the penalized component of each smooth is treated as a random effect term,
while the unpenalized component is treated as fixed. The wiggliness penalty matrix for the smooth is in effect the
precision matrix when the smooth is treated as a random effect. Estimating the degree of smoothness of the term
amounts to estimating the variance parameter for the term.
gamm4 uses the same reparameterization trick employed by gamm to allow any single quadratic
penalty smoother to be used (see Wood, 2004, or 2006 for details). Given the reparameterization then the modular fitting approach employed in lmer can be used to fit a GAMM. Estimation is by
Maximum Likelihood in the generalized case, and REML in the gaussian additive model case. gamm4 allows
the random effects specifiable with lmer to be combined with any number of any of the (single penalty) smooth
terms available in gam from package mgcv as well as t2 tensor product smooths.
Note that the model comparison on the basis of the (Laplace
approximate) log likelihood is possible with GAMMs fitted by gamm4.
As in gamm the smooth estimates are assumed to be of interest, and a covariance matrix is returned which
enables Bayesian credible intervals for the smooths to be constructed, which treat all the terms in random as random.
For details on how to condition smooths on factors, set up varying coefficient models, do signal regression or set up terms
involving linear functionals of smooths, see gam.models, but note that te type tensor product and adaptive smooths are
not available with gamm4.
Value
Returns a list with two items:
gam |
an object of class |
mer |
the fitted model object returned by |
WARNINGS
If you don't need random effects in addition to the smooths, then gam is substantially faster, gives fewer convergence warnings, and slightly better MSE performance (based on simulations).
Models must contain at least one random effect: either a smooth with non-zero
smoothing parameter, or a random effect specified in argument random.
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.
Linked smoothing parameters, adaptive smoothing and te terms are not supported.
This routine is obviously less well tested than gamm.
Author(s)
Simon N. Wood simon.wood@r-project.org
References
Bates D., M. Maechler, B. Bolker & S. Walker (2013). lme4: Linear mixed-effects models using Eigen and S4. https://cran.r-project.org/package=lme4
Wood S.N., Scheipl, F. and Faraway, J.J. (2013/2011 online) Straightforward intermediate rank tensor product smoothing in mixed models. Statistics and Computing 23(3): 341-360
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. (2006) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press.
For more GAMM references see gamm
http://www.maths.bris.ac.uk/~sw15190/
See Also
gam, gamm, gam.models, lmer, predict.gam, plot.gam, summary.gam, s, vis.gam
Examples
## NOTE: most examples are flagged as 'do not run' simply to
## save time in package checking on CRAN.
###################################
## A simple additive mixed model...
###################################
library(gamm4)
set.seed(0)
dat <- gamSim(1,n=400,scale=2) ## simulate 4 term additive truth
## Now add 20 level random effect `fac'...
dat$fac <- fac <- as.factor(sample(1:20,400,replace=TRUE))
dat$y <- dat$y + model.matrix(~fac-1)%*%rnorm(20)*.5
br <- gamm4(y~s(x0)+x1+s(x2),data=dat,random=~(1|fac))
plot(br$gam,pages=1)
summary(br$gam) ## summary of gam
summary(br$mer) ## underlying mixed model
anova(br$gam)
## compare gam fit of the same
bg <- gam(y~s(x0)+x1+s(x2)+s(fac,bs="re"),
data=dat,method="REML")
plot(bg,pages=1)
gam.vcomp(bg)
##########################
## Poisson example GAMM...
##########################
## simulate data...
x <- runif(100)
fac <- sample(1:20,100,replace=TRUE)
eta <- x^2*3 + fac/20; fac <- as.factor(fac)
y <- rpois(100,exp(eta))
## fit model and examine it...
bp <- gamm4(y~s(x),family=poisson,random=~(1|fac))
plot(bp$gam)
bp$mer
## Not run:
#################################################################
## Add a factor to the linear predictor, to be modelled as random
## and make response Poisson. Again compare `gamm' and `gamm4'
#################################################################
set.seed(6)
dat <- gamSim(1,n=400,scale=2) ## simulate 4 term additive truth
## add random effect...
g <- as.factor(sample(1:20,400,replace=TRUE))
dat$f <- dat$f + model.matrix(~ g-1)%*%rnorm(20)*2
dat$y <- rpois(400,exp(dat$f/7+1))
b2<-gamm(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson,
data=dat,random=list(g=~1))
plot(b2$gam,pages=1)
b2r<-gamm4(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson,
data=dat,random = ~ (1|g))
plot(b2r$gam,pages=1)
rm(dat)
vis.gam(b2r$gam,theta=35)
##################################
# Multivariate varying coefficient
# With crossed and nested random
# effects.
##################################
## Start by simulating data...
f0 <- 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))
}
f1 <- function(x2) 2 * sin(pi * x2)
f2 <- function(x2) exp(2 * x2) - 3.75887
f3 <- function (x2) 0.2 * x2^11 * (10 * (1 - x2))^6 + 10 * (10 * x2)^3 *
(1 - x2)^10
n <- 1000
## first set up a continuous-within-group effect...
g <- factor(sample(1:50,n,replace=TRUE)) ## grouping factor
x <- runif(n) ## continuous covariate
X <- model.matrix(~g-1)
mu <- X%*%rnorm(50)*.5 + (x*X)%*%rnorm(50)
## now add nested factors...
a <- factor(rep(1:20,rep(50,20)))
b <- factor(rep(rep(1:25,rep(2,25)),rep(20,50)))
Xa <- model.matrix(~a-1)
Xb <- model.matrix(~a/b-a-1)
mu <- mu + Xa%*%rnorm(20) + Xb%*%rnorm(500)*.5
## finally simulate the smooth terms
v <- runif(n);w <- runif(n);z <- runif(n)
r <- runif(n)
mu <- mu + f0(v,w)*z*10 + f3(r)
y <- mu + rnorm(n)*2 ## response data
## First compare gamm and gamm4 on a reduced model
br <- gamm4(y ~ s(v,w,by=z) + s(r,k=20,bs="cr"),random = ~ (1|a/b))
ba <- gamm(y ~ s(v,w,by=z) + s(r,k=20,bs="cr"),random = list(a=~1,b=~1),method="REML")
par(mfrow=c(2,2))
plot(br$gam)
plot(ba$gam)
## now fit the full model
br <- gamm4(y ~ s(v,w,by=z) + s(r,k=20,bs="cr"),random = ~ (x+0|g) + (1|g) + (1|a/b))
br$mer
br$gam
plot(br$gam)
## try a Poisson example, based on the same linear predictor...
lp <- mu/5
y <- rpois(exp(lp),exp(lp)) ## simulated response
## again compare gamm and gamm4 on reduced model
br <- gamm4(y ~ s(v,w,by=z) + s(r,k=20,bs="cr"),family=poisson,random = ~ (1|a/b))
ba <- gamm(y ~ s(v,w,by=z) + s(r,k=20,bs="cr"),family=poisson,random = list(a=~1,b=~1))
par(mfrow=c(2,2))
plot(br$gam)
plot(ba$gam)
## and now fit full version (very slow)...
br <- gamm4(y ~ s(v,w,by=z) + s(r,k=20,bs="cr"),family=poisson,random = ~ (x|g) + (1|a/b))
br$mer
br$gam
plot(br$gam)
####################################
# Different smooths of x2 depending
# on factor `fac'...
####################################
dat <- gamSim(4)
br <- gamm4(y ~ fac+s(x2,by=fac)+s(x0),data=dat)
plot(br$gam,pages=1)
summary(br$gam)
####################################
# Timing comparison with `gam'... #
####################################
dat <- gamSim(1,n=600,dist="binary",scale=.33)
system.time(lr.fit0 <- gam(y~s(x0)+s(x1)+s(x2),
family=binomial,data=dat,method="ML"))
system.time(lr.fit <- gamm4(y~s(x0)+s(x1)+s(x2),
family=binomial,data=dat))
lr.fit0;lr.fit$gam
cor(fitted(lr.fit0),fitted(lr.fit$gam))
## plot model components with truth overlaid in red
op <- par(mfrow=c(2,2))
fn <- c("f0","f1","f2","f3");xn <- c("x0","x1","x2","x3")
for (k in 1:3) {
plot(lr.fit$gam,select=k)
ff <- dat[[fn[k]]];xx <- dat[[xn[k]]]
ind <- sort.int(xx,index.return=TRUE)$ix
lines(xx[ind],(ff-mean(ff))[ind]*.33,col=2)
}
par(op)
## End(Not run)
######################################
## A "signal" regression example, in
## which a univariate response depends
## on functional predictors.
######################################
## simulate data first....
rf <- function(x=seq(0,1,length=100)) {
## generates random functions...
m <- ceiling(runif(1)*5) ## number of components
f <- x*0;
mu <- runif(m,min(x),max(x));sig <- (runif(m)+.5)*(max(x)-min(x))/10
for (i in 1:m) f <- f+ dnorm(x,mu[i],sig[i])
f
}
x <- seq(0,1,length=100) ## evaluation points
## example functional predictors...
par(mfrow=c(3,3));for (i in 1:9) plot(x,rf(x),type="l",xlab="x")
## simulate 200 functions and store in rows of L...
L <- matrix(NA,200,100)
for (i in 1:200) L[i,] <- rf() ## simulate the functional predictors
f2 <- function(x) { ## the coefficient function
(0.2*x^11*(10*(1-x))^6+10*(10*x)^3*(1-x)^10)/10
}
f <- f2(x) ## the true coefficient function
y <- L%*%f + rnorm(200)*20 ## simulated response data
## Now fit the model E(y) = L%*%f(x) where f is a smooth function.
## The summation convention is used to evaluate smooth at each value
## in matrix X to get matrix F, say. Then rowSum(L*F) gives E(y).
## create matrix of eval points for each function. Note that
## `smoothCon' is smart and will recognize the duplication...
X <- matrix(x,200,100,byrow=TRUE)
## compare `gam' and `gamm4' this time
b <- gam(y~s(X,by=L,k=20),method="REML")
br <- gamm4(y~s(X,by=L,k=20))
par(mfrow=c(2,1))
plot(b,shade=TRUE);lines(x,f,col=2)
plot(br$gam,shade=TRUE);lines(x,f,col=2)