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RE: st: logarithmic scales


From   Roger Newson <roger.newson@kcl.ac.uk>
To   statalist@hsphsun2.harvard.edu
Subject   RE: st: logarithmic scales
Date   Mon, 01 Dec 2003 16:42:36 +0000

At 15:31 01/12/03 +0000, Nick Cox wrote:

I'd assert, perhaps very rashly, that beyond
some threshold, very low P-values are
practically indistinguishable. I suppose that
log P-value of -20 is often appealing as a kind of
thermonuclear demolition of a null hypothesis, but I wonder
if anyone would think differently of (say) -6. Also,
as is well known, the further you go out into
the tail the more you depend on everything being
as it be (model assumptions, data without
measurement error, numerical analysis...).
On the other hand, there are situations
in which an overwhelming P-value is needed
for any ensuing decision.
A good discussion of this issue is given in Subsection 35.7 of Kirkwood and Sterne (2003), which is a basic text aimed mostly at non-mathematicians. This uses a Bayesian heuristic, based on the well-known result that the posterior odds between 2 hypotheses after the data analysis is equal to the prior odds between the same 2 hypotheses multiplied by the likelihood ratio between the 2 hypotheses. It is argued that a P-value below 0.003 is good enough for most of the people most of the time, because, *if* the prior odds are as bad as 100:1 against a nonzero population difference, *and* the power to detect a difference significant at P<=0.001 is as low as 0.5, *then* the posterior odds in favour of a nonzero population difference, given a P-value <=0.001, will be 5:1 in favour.

This heuristic seems to make sense to me, if the P-value is for the parameter of prior interest in the study design protocol, because not many grant-awarding bodies will pay for a study for which they consider the prior odds of an interesting difference to be worse than 100:1 against. On the other hand, in the real world, with today's technology, it is nearly always cheaper to torture the data until they confess than to collect more data. Therefore, a lot of people's colleagues expect them to do "subset analyses from hell", and are reluctant to write up negative results as such. Therefore, an honest scientist who wants to accumulate publications is often not a data miner, but a "data lawyer", cross-examoining the data on the moral equivalent of a "no-win no-fee" contract. Under these conditions, a lot of statistically-minded scientists will forget what they learned at college, and do what they are told, and torture the data. If the P-value is from one of a sequence of subset analyses, and is undertaken posterior to a main analysis which found nothing, then, arguably, the "prior odds" against an interesting difference might reasonably be worse than 100:1 against.

Roger

References

Kirkwood BR, Sterne JAC. Essential medical statistics. Second edition. Oxford, UK: Blackwell Science; 2003.


--
Roger Newson
Lecturer in Medical Statistics
Department of Public Health Sciences
King's College London
5th Floor, Capital House
42 Weston Street
London SE1 3QD
United Kingdom

Tel: 020 7848 6648 International +44 20 7848 6648
Fax: 020 7848 6620 International +44 20 7848 6620
or 020 7848 6605 International +44 20 7848 6605
Email: roger.newson@kcl.ac.uk
Website: http://www.kcl-phs.org.uk/rogernewson

Opinions expressed are those of the author, not the institution.

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