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From |
"FEIVESON, ALAN H. (AL) (JSC-SK) (NASA)" <alan.h.feiveson@nasa.gov> |

To |
"'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu> |

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

Date |
Fri, 19 Mar 2004 10:26:19 -0600 |

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: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Nick Cox Sent: Friday, March 19, 2004 9:23 AM To: statalist@hsphsun2.harvard.edu 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 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. > * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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