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Re: st: Binomial confidence intervals
I disagree. This is too drastic a summary. If this were all there is
and you prefer A, then I have a terrific CI for you: It satisfies A, it
is robust to everything, including the data, and it is the interval zero
to one. It is guaranteed to work every time.
No, there is more to the controversy, including the length of the
interval, and, some would add, the center of the interval.
I agree with the authors of the paper when they say that approximation
plays a central role to the whole discussion--
including why the interval zero to one is silly!
The controversy can be summarized like this:
A. Suppose I have a true probability of success P that I am trying to
estimate with a sample of size N. If I draw 1 million samples and
compute my interval, I want at least 95% of those to cover the true P.
I want this to happen, irrespective of the true value of P and the
sample size N (ie, 100% of the time). So, if you have a different
situation with probability P* and sample size N*, you should also have
coverage of 95% or better. In other words, this means that I want an
interval with coverage AT LEAST 95% ALL THE TIME (for some situations,
it will have more).
B. In contrast, I may want the 95% coverage to be "on average," across
different situations. In other words, in some situation of P and N, I
am willing to accept coverage less than 95%, while for some other
situation P* and N*, I will have coverage better than 95%. All I want
is my interval to give at least 95% coverage ON AVERAGE.
(A) leads us to conservative intervals such as Clopper-Pearson. They
have to cover the truth 95% of the time no matter what the situation.
Even a single situation where they fall to 94.9% is not acceptable.
However, they need not be as conservative as Clopper-Pearson. The
Blyth-Still-Casella is much better (ie, it stays at 95% or better no
matter what, but does not become as conservative as Clopper-Pearson)
and is very competitive with those advocated in Agresti & Coull and
Brown et al.
(B) leads us to generally shorter intervals, but in some situations
they fall below 95% (sometimes way below that). Brown et al. propose
certain "corrections" to remove the most severe dips in coverage
probability. Still, none of those intervals is guaranteed to have 95%
coverage or better in
a particular situation. And the problem is we cannot be sure how much
below 95% they can fall. All we know is that they will have 95%
coverage on average.
I go with A. I think that B is a bit ad hoc and a dip below 95% may be
acceptable to me but unacceptable to you. But either approach seems
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Constantine Daskalakis, ScD
Biostatistics Section, Thomas Jefferson University,
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