FAQ
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...

Alan T. Arnholt
Associate Professor
Dept. of Mathematical Sciences
Appalachian State University

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  • Peter Dalgaard at Dec 29, 2005 at 8:11 pm

    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.

    --
    O__ ---- Peter Dalgaard ??ster Farimagsgade 5, Entr.B
    c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K
    (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
    ~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk) FAX: (+45) 35327907

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