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#### Options  Subscribe or Unsubscribe   Log In   Get Password Subject: Re: Distribution of point estimates within confidence intervals

From:  Date: Mon, 27 Mar 2006 11:08:53 +0100

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 ```On 27-Mar-06 Andrew Jull wrote: > In 'User's Guides to the Medical Literature' the fine book > edited by Guyatt and Rennie it is noted that values further > away from the point estimate are less probable than those > closer to the point estimate eg. With a 95%CI 1-16%, > the extreme values in the confidence interval are less likely > to occur than those around the observed point estimate eg 8%. > I have seen this stated elsewhere, and while it makes sense > intuitively I have not seen any evidence to support this view. > Can anyone direct me to the evidence? Paul Glasziou has commented from a Bayesiam viewpoint, and correctly from that viewpoint. However, that is a quite different ball-game from the basis on which a Confidence Interval rests. The principle of confidence interval estimation is that, in repeated sampling from the same population, followed in each case by calculating the (say 95%) CI, a series of random intervals emerges, which have the property that the probability is 95% that such an interval will embrace the parameter value being estimated. Therefore, when (as is usually the case) you calculate a CI from a single sample and assert that it includes the value being estimated, you are betting (at odds of 19:1) that your statement is true. If you want better odds (e.g. 99:1) then choose a higher confidence level (99%). (Indeed, to understand the nature of the CI, imagine a horse-race where you can bet on a horse and then watch the race on CCTV -- except that the only horse the system allows you to see is the one you have bet on, the others being rendered invisible by the system. Then you see your horse cross the line at the end of the race. Did it win? Well, if the odds (say 19:1) are an accurate representation of the probability that it should win, in such a race, then you can be 95% confident that it did win. Now go and argue with the bookie.) So the situation with a CI is that the unknown quantity being stimated is fixed, and the CI varies randomly from sample to sample and will include the true value in a due proportion of cases. Now it is also the case that the more extreme CIs (i.e. those that are relatively off to one side of the true value) will be rarer than the less extreme ones; and indeed you can attach a "distribution" to this by considering what happens as you enhance the confidence level -- e.g. from 95% to 96% -- which will give rise to 95% CIs each of which will include a 95% CI. The extra bits at either end are the extensions of the 95% CI arising from the change from 95% to 96%; the lengths of these extensions are greater, the higher the confidence level; and so can be anvisioned as corresponding to lower "density". However, looking at it in this way presupposes that there is something random going on which possesses this probability "distribution". In fact, this is the random confidence interval itself, and not the unknown parameter being estimated which, in confidence interval terms, is a fixed (so not random) quantity. In order for the parameter itself to be viewed as random, which is what is implied by the kind of query you have extressed above, it needs to be endowed with a probability distribution. This is where the Bayesian viewpoint comes in. To arrive at a Bayesian conclusion, it is both necessary and sufficient that the parameter has a *prior* probability distribution; then, once the data have been obtained, Bayes's theorem can be used to derive a *posterior* probability distribution. From this you can then calculate the "credible intervals" discussed by Paul Glasziou. It is often the case that the interval obtained by a Confidence Interval method is numerically the same as the credible interval obtained by a Bayesian method using a particular prior distribution. However, the interpretation is quite different (as described above). In particular, part of the rationale for the CI approach is that the probability that the statement will be true is independent of any assumption about a Bayesian prior; while a Bayesian postieror statement will depend on the choice of prior. > Also, does it follow that if a trial is repeated X times, > the observed point estimate for each trial be normally distributed > within the confidence interval of an adequately powered trial? No. First of all, an "adequately powered" trial may be so much adequately powered that the CI for that trial is very short indeed. However, the distribution of the point estimate for a trial differently powered may be scattered ofver a much wider range. Secondly, suppose the point-estimate methodology is unbiased. Now take the CI for the "adequately powered trial". The "midpoint" will be the point estimate from that trial. Even if the methodology is unbiased, that particular value will have an error. So the particular CI you have obtained from that trial will be offset relatively to the true value of the parameter. Hence, when you embark on the X repetitions, you will obtain a series of estimates which, overall, are unbiasedly distributed relative to the true value. Hence their distribution will be offset relative to the CI you obtained from the "adequately powered trial". Hoping this helps to clarify ... Best wishes, Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <[log in to unmask]> Fax-to-email: +44 (0)870 094 0861 Date: 27-Mar-06 Time: 11:08:48 ------------------------------ XFMail ------------------------------ ```