Re: [R] use of predict() with confidence/prediction bands

From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>
Date: Fri 30 Dec 2005 - 07:11:36 EST

Alan Arnholt <arnholt@cs.cs.appstate.edu> writes:

> To my understanding, a confidence interval typically covers a single
> valued parameter. In contrast, a confidence band covers an entire line
> with a band. In regression, it is quite common to construct confidence
> and prediction bands. I have found that many people are connecting
> individual confidence/prediction interval values produced with
> predict(object,sd.fit=T,type="conf/pred") and calling the result a
> confidence/prediction band. Since there is no specific probability
> statement that can be attached to these connected confidence/prediction
> intervals, this does not seem reasonable to me. This is done, for
> example, in ISWR pg. 105, UsingR for Introductory Statistics pg 296, and
> Linear Models with R pg. 39 (Although in this instance the intervals are
> called 95% "pointwise" confidence bands versus simply 95% confidence
> bands.) To make a confidence/prediction band, one should construct
> simultaneous confidence/prediction intervals with say a Scheffe approach
> as mentioned in the S-PLUS Guide to statistics pg 274. If these connected
> intervals were called pointwise confidence/prediction intervals with the
> understanding that have no particular probability interpretation, then
> they are useful in understanding where the line should fall. However,
> they are not confidence/prediction bands as such, and I think it is
> misleading to name them so. Should the intervals the authors in the
> three mentioned references construct not be called something similar
> to connected 95% pointwise confidence/prediction intervals versus 95%
> confidence/prediction bands? Or, have I missed the boat? Fire away...

You do have a point, of course. My take is that (a) they are bands and (b) they have the property that for _each_ x they contain y(x) with 95% probability. So "95% pointwise confidence bands" is reasonably warranted to my mind. ISwR could probably be more careful in making the "pointwise" distinction, but I'd be afraid of confusing readers who might well be at the level where the prime difficulty is grasping the difference between prediction intervals and confidence intervals.

Global coverage, i.e., bands that contain the true line with 95% probability, is quite a bit harder to obtain, especially in the nonparametric regression extensions. Such bands end up being rather wide, and some (I'm afraid I forgot who) have suggested just to use the pointwise bands with the understanding that they cover, on average, 95% of the true line.

-- 
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  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
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Received on Fri Dec 30 07:17:10 2005

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