One place where tiny p-values are important is in multiple-comparison tests,
where you're applying some sort of Bonferroni-like correction or venturing
into False Discovery rates and the like. If you stack enough tests on top
of one another in a given analysis, its likely you'll meet that p-value
cutoff of significance purely by chance...
I've seen a fair number of presentations (powerpoint-"disabled" to boot)
with lecturers implying the thermonuclear detonation of the null hypothesis
(thanks for the image, Nick). It seems to me that in such instances rather
than going after the null with such bloodthirsty vigour a better use of time
would be spent looking at the alternative hypotheses (i.e., the variable
accounts for no more than N% of the variance in the population measured)
From: Roger Newson [mailto:firstname.lastname@example.org]
Sent: Monday, December 01, 2003 8:43 AM
Subject: RE: st: logarithmic scales
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.
Kirkwood BR, Sterne JAC. Essential medical statistics. Second edition.
Oxford, UK: Blackwell Science; 2003.
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