# 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

1 |

### Arguments

`x` |
a fitted model object of class |

`rescale` |
the penalty matrices for smooths are rescaled before fitting, for numerical
stability reasons, if |

`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. |

### 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.

### Value

Either a vector of variance components for each smooth term (as standard deviations), or a matrix. The first column of the matrix gives standard deviations for each term, while the subsequent columns give lower and upper confidence bounds, on the same scale.

For models in which there are more smoothing parameters than actually estimated (e.g. if some were fixed, or smoothing parameters are linked) then a list is returned. The `vc`

element is as above, the `all`

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

The routine prints a table of estimated standard deviations and confidence limits, if these can be computed, and reports the numerical rank of the covariance matrix.

### Author(s)

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

### 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | ```
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)
``` |

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