Garry - In order to construct an exact confidence interval using classical
methods, the probability of observing a difference as extreme or more
extreme than the one you got, would have to be a monotone function of the
difference betwen the "true" probabilities of occurrence. But this is not
the case - the above probability is not an explicit function of the
difference alone; it also depends on the actual values of the "true"
probabilities. A better approach would be to use Bayesian methods. Then you
could get a posterior distribution of the difference that would make "sense"
(i.e it would have support only in [-1, +1] ).
Al Feiveson
-----Original Message-----
From: [email protected]
[mailto:[email protected]]On Behalf Of Nick Cox
Sent: Friday, March 19, 2004 9:23 AM
To: [email protected]
Subject: RE: st:Confidence interval of difference between two
proportions and -csi-
The terminology "exact" is indeed used in this way, and there's
scarcely a chance of changing that terminology.
But as a matter of ordinary English it's potentially highly
misleading term for anyone who prefers that (for example)
95% means precisely that. I guess for everyone who's ingested
this explanation there are many more who think in terms of
coverage (without necessarily using that term).
Nick
[email protected]
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]]On Behalf Of Dupont,
> William
> Sent: 19 March 2004 15:09
> To: [email protected]
> Subject: RE: st:Confidence interval of difference between two
> proportions and -csi-
>
>
> Statalisters
>
> I believe that there is some confusion about the meaning of exact
> confidence intervals. Confidence intervals are defined in two ways.
> Let theta be a parameter and L U be two statistics. Then confidence
> intervals are defined as follows:
>
> Coverage definition:
>
> (L, U) is a 95% confidence interval for theta if Pr[L < theta < U] =
> 0.95
>
> Non-rejection definition:
>
> A 95% confidence interval, (L, U), consists of all values of
> theta that
> can not be rejected at the 5% significance level given the data.
>
> These two definitions are equivalent for normally distributed data in
> which the mean and variance are unrelated. In epidemiology and other
> disciplines we often work with statistics (e.g. odds ratios) in which
> these definitions yield different intervals. Exact
> confidence intervals
> use the non-rejection definition. When estimating odds
> ratios from 2x2
> tables, the total number of successes in both groups is
> close to being
> an ancillary statistic in the sense that knowing this total tells us
> nothing about the true odds ratio. The Conditionality
> Principle requires
> that we condition our inferences on ancillary statistics. It is for
> this reason that we condition on the marginal totals of a 2x2
> table when
> making inferences about odds ratios.
>
> If you accept the conditionality argument then the usual exact
> confidence interval is correctly derived from the hypergeometric
> distribution. It is an exact interval not because it uses the
> hypergeometric distribution but because it complies with the
> non-rejection definition given above. It should be noted that when
> these definitions disagree, the non-rejection confidence interval will
> have a higher coverage probability than the analogous
> interval obtained
> by the coverage definition. In this sense, it is a more conservative
> interval.
>
> See Rothman and Greenland (1998) for further details.
>
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