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
