The Agresti-Coull CI's are based on the counting variance in a binomial
process while the naive CI's are (presumably? Maybe several methods are
refered to as "naive") based on the variance across the proportions only.
So the two are very different concepts and not comparable at all.
Say you have two proportions, namely
Since the denominator (one million) is so large that the counting variance
can be ignored, the two methods would (roughly) agree (here the SD happens
to be zero so the naive CI collapses but the A-C ci will be very small
But if your two proportions are based on small denominators, say
you will have a huge width of the A-C CI because of the large counting
variance. Yet the naive CI will again collapse because the SD of (2/4 ,
2/4) is zero.
The A-C may be more accurate than the naive CI but the modeling assumptions
I have computed the Agresti Coull CI's for proportions along with the
> naive CI's for proportions and the naive CI's for means (since, in my
> case, the proportions in question are really mean proportions, both
> approaches seem reasonable).
> However, I have obtained CI's with wildly differing widths among these
> approaches (holding confidence level, and sample size constant).
> Specifically, the Agresti Coull CI's are 3 to 4 times wider than the
> naive (unbounded) ones. I'm having a hard time convincing myself that
> they both provide 95% coverage.
> Any explanations or insight?
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