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On 20-May-99 Stephen Senn wrote: > An infinity problem arises commonly in the application of Fieller's > theorem for the confidence interval for a ratio (or a "fiducial > interval" as Fieller would have described it) and there has been much > debate in the pharmaco-economic literature regarding cost benefit > ratios. Again, in my view, a lot of the difficulty comes from trying to > produce simple summaries of what are of necessity complex > questions. The "infinity problem" is less obscure than it is sometimes made to seem. For a simple example, consider a regression estimated from data pairs (Xi,Yi): y = a + b*x is estimated as a_hat + b_hat*x. Now consider estimating the value x0 of x for which y has a specific value y0: y0 = a + b*x0. Try x0_hat = (y0 - a_hat)/b_hat. This has a form to which Fieller's theorem can be applied, and infinite confidence intervals for x0 can arise. This looks like an obscure consequence of theory difficult to understand. However, suppose the data do not, at significance level 0.05 say, suffice to reject the hypothesis "b=0" that the slope is zero. Then +/- infinity is not a rejected possible value of x0, and the 95% confidence interval for x0 will be infinite. It is as simple as that. If the same data suffice to reject "b=0" at significance level 0.10, then the 90% CI for x0 will be finite. The maximum confidence with which you can assert that x0 lies within any finite interval is 1 minus the minimum significance level at which "b=0" is rejected. The basic truth of the matter is that the data are inadequate to support any greater confidence. Similarly, if in comparing two proportions the minimum significance level at which "p1 = p2" can be rejected is, say, alpha, then the maximum confidence with which any informative (i.e. non-infinite) range for NNT canbe asserted is P = 1 - alpha. The data are inadequate to support any greater level of confidence. Turning the issue round from "the 95% CI is infinite", which looks puzzling, to "the data do not support 95% confidence", which is very straightforward, is the truth of the matter. Best wishes to all, Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <[log in to unmask]> Date: 21-May-99 Time: 00:07:13 ------------------------------ XFMail ------------------------------ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%