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

[Peter Dalgaard]

Alan Arnholt <arnholt at 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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