mgcv (version 1.9-4)

gam.vcomp: Report gam smoothness estimates as variance components

Description

GAMs can be viewed as mixed models, where the smoothing parameters are related to variance components. This routine extracts the estimated variance components associated with each smooth term, and if possible returns confidence intervals on the standard deviation scale.

Usage

gam.vcomp(x,rescale=TRUE,conf.lev=.95)
# S3 method for gam.vcomp
print(x,...)

Value

Either a vector of variance components for each smooth term (as standard deviations), or a list. A list will be returned if the smoothness selection method used was REML or ML, or where a model has more smoothing parameters than actually estimated.

If a list, it will contain some or all of the elements described depending on the method of smoothness selection used, and whether some smoothing parameters were not estimated.

Depending on the smoothness selection method, element vc may contain a matrix or a vector. If REML or ML smoothness selection was used,

vc will be a matrix whose first column gives standard deviations for each term, while the subsequent columns give lower and upper confidence bounds, on the same scale. For models fitted with other smoothness selection methods, component vc will be a vector of standard deviations.

The all element is a vector of variance components for all the smoothing parameters (estimated + fixed or replicated).

Additionally, for REML or ML smoothness selection, the numerical rank of the covariance matrix and the Hessian matrix are returns as elements rank

and rank.hess respectively.

Arguments

x

a fitted model object of class gam as produced by gam(), or an onject of class gam.vcomp for the print method.

rescale

the penalty matrices for smooths are rescaled before fitting, for numerical stability reasons, if TRUE this rescaling is reversed, so that the variance components are on the original scale.

conf.lev

when the smoothing parameters are estimated by REML or ML, then confidence intervals for the variance components can be obtained from large sample likelihood results. This gives the confidence level to work at.

...

other arguments.

Author

Simon N. Wood modified by Gavin Simpson

Details

The (pseudo) inverse of the penalty matrix penalizing a term is proportional to the covariance matrix of the term's coefficients, when these are viewed as random. For single penalty smooths, it is possible to compute the variance component for the smooth (which multiplies the inverse penalty matrix to obtain the covariance matrix of the smooth's coefficients). This variance component is given by the scale parameter divided by the smoothing parameter.

This routine computes such variance components, for gam models, and associated confidence intervals, if smoothing parameter estimation was likelihood based. Note that variance components are also returned for tensor product smooths, but that their interpretation is not so straightforward.

The routine is particularly useful for model fitted by gam in which random effects have been incorporated.

References

Wood, S.N. (2008) Fast stable direct fitting and smoothness selection for generalized additive models. Journal of the Royal Statistical Society (B) 70(3):495-518

Wood, S.N. (2011) Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society (B) 73(1):3-36

See Also

smooth.construct.re.smooth.spec

Examples

  set.seed(3) 
  require(mgcv)
  ## simulate some data, consisting of a smooth truth + random effects

  dat <- gamSim(1,n=400,dist="normal",scale=2)
  a <- factor(sample(1:10,400,replace=TRUE))
  b <- factor(sample(1:7,400,replace=TRUE))
  Xa <- model.matrix(~a-1)    ## random main effects
  Xb <-  model.matrix(~b-1)
  Xab <- model.matrix(~a:b-1) ## random interaction
  dat$y <- dat$y + Xa%*%rnorm(10)*.5 + 
           Xb%*%rnorm(7)*.3 + Xab%*%rnorm(70)*.7
  dat$a <- a;dat$b <- b

  ## Fit the model using "re" terms, and smoother linkage  
  
  mod <- gam(y~s(a,bs="re")+s(b,bs="re")+s(a,b,bs="re")+s(x0,id=1)+s(x1,id=1)+
               s(x2,k=15)+s(x3),data=dat,method="ML")

  gam.vcomp(mod)