[R]gam and concurvity

[Simon Wood]

> in the paper "Avoiding the effects of concurvity in GAM's .." of
> Figueiras et al. (2003) it is mentioned that in GLM collinearity is taken 
> into account in the calc of se but not in GAM (-> results in confidence 
> interval too narrow, p-value understated,  GAM S-Plus version). I haven't 
> found any references to GAM and concurvity or collinearity on the R page. 
> And I wonder if the R  version of Gam differ in this point.

- the penalized regression spline representation means that it's easy to
calculate the `correct' s.e.'s and this is what is done. The covariance
matrix used is based on a Bayesian model of smoothing, generalized from
Silverman (1985), JRSSB (and less closely, Wahba, 1983, JRSSB), so the
s.e.'s are generally a little larger than you'd get if you just pretended
that the GAM was an un-penalized GLM (this widening generally improves CI
performance). 

As Thomas Lumley pointed out, the s.e.'s don't take into account smoothing
parameter estimation uncertainty. In simulation studies this
uncertainty seems to have very little effect on the realized coverage
probabilities of Confidence Interval's that are in some sense `whole
model' intervals, but the performance of CI's for component functions of
the GAM can be quite a long way from nominal. There's a simple
`not-very-computer-intensive' fix for this which removes the conditioning
on the smoothing parameters and greatly improves component-wise coverage
probabilities.... implementation is on my `to-do' list (might wait to see
what the referees say though!)

Simon 

ps. mgcv 0.9 out now! (changes list linked to my www page)
_____________________________________________________________________
> Simon Wood simon at stats.gla.ac.uk        www.stats.gla.ac.uk/~simon/
>>  Department of Statistics, University of Glasgow, Glasgow, G12 8QQ
>>>   Direct telephone: (0)141 330 4530          Fax: (0)141 330 4814