gamm                  package:mgcv                  R Documentation

_G_e_n_e_r_a_l_i_z_e_d _A_d_d_i_t_i_v_e _M_i_x_e_d _M_o_d_e_l_s

_D_e_s_c_r_i_p_t_i_o_n:

     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'.

_U_s_a_g_e:

     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,...)

_A_r_g_u_m_e_n_t_s:

 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'

_D_e_t_a_i_l_s:

     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'.

_V_a_l_u_e:

     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.

_W_A_R_N_I_N_G_S:

     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.

_A_u_t_h_o_r(_s):

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

_R_e_f_e_r_e_n_c_e_s:

     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

     <URL: http://www.stats.gla.ac.uk/~simon/>

_S_e_e _A_l_s_o:

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

_E_x_a_m_p_l_e_s:

     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)

