[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
Re: st: Binomial confidence intervals
At 08:03 AM 9/8/2004, Richard Williams wrote:
The controversy can be summarized like this:
At 11:47 AM 9/8/2004 +0100, Paul Seed wrote:
A few key quotes:
As I don't have access to a decent Stats library here, I tried to obtain
the recommended paper (Brown, Cai, & DasGupta.
Interval Estimation for a Binomial Proportion. Statistical Science, 2001,
16, pp. 101-133.) over the internet; but it is currently behind a "rolling
firewall", until 2005.
p. 115 - "Based on this analysis, we recommend the Wilson or the
equal-tailed Jeffreys prior interval for small n (n is less than or equal
to 40). These two intervals are comparable in both absolute error and
length for n is less than or equal to 40, and we believe that either could
be used, depending on taste."
p. 115 - "For larger n, the Wilson, the Jeffreys and the AgrestiCoull
intervals are all comparable, and the AgrestiCoull interval is the
simplest to present....we recommend the AgrestiCoull interval for
practical use when n is greater than or equal to 40. Even for small sample
sizes, the easy-to-present AgrestiCoull interval is much preferable to
the standard one."
p. 113 - "The ClopperPearson interval is wastefully conservative and is
not a good choice for practical use."
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 quite
The documents accompanying this transmission may contain confidential
health or business information. This information is intended for the use of
the individual or entity named above. If you have received this information
in error, please notify the sender immediately and arrange for the return
or destruction of these documents.
Constantine Daskalakis, ScD
Biostatistics Section, Thomas Jefferson University,
211 S. 9th St. #602, Philadelphia, PA 19107
* For searches and help try: