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RE: st:Confidence interval of difference between two proportions and -csi-


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st:Confidence interval of difference between two proportions and -csi-
Date   Fri, 19 Mar 2004 15:22:49 -0000

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 
n.j.cox@durham.ac.uk 

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu
> [mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Dupont,
> William
> Sent: 19 March 2004 15:09
> To: statalist@hsphsun2.harvard.edu
> 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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