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.

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Date: 27-Mar-06                                       Time: 11:08:48
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