More formally consider the following. Under Ho a test
statistic x might have a distribution with density f(x) and distribution
function F(x) -- could be normal but need not be.
The p-value for a one sided test associated with a sample outcome x is
F(x) --
It is then fairly easy to show with a bit of integral calculus on
transformation of variables that the distribution of p=F(x) is uniform on
(0,1) - the argument can be extended quite easily for two sided tests.
I think I first came accross this result as a practical exercise when I
first learned distribution theory as an undergraduate
Tony Fielding
At 17:03 05/11/2007, you wrote:
>The statement " all p-values between 0 and 1 are equally likely" is just
>another way of saying that under H0:
>
>"the chance of getting p<.01 is 0.01;
>the chance of getting p<.05 is 0.05;
>in fact the chance of getting p<x% is always x%"
>
>JOHN BIBBY
>
>
>
>
>-----Original Message-----
>From: A UK-based worldwide e-mail broadcast system mailing list
>[mailto:[log in to unmask]] On Behalf Of kornbrot
>Sent: 05 November 2007 14:40
>To: [log in to unmask]
>Subject: Query
>
>An electronic text makes the following assertion:
>
>When the null hypothesis, H0, is true, all p-values between 0 and 1 are
>equally likely. In other words, the p-value has a rectangular distribution
>between 0 and 1.
>
>Is that so? I would have thought that if the null is true then, the test
>statistic is [asymtotically] normally distiributed about its mean, and
>p=values near to .5 would be more probable than extreme p-values.
>
>Can anyone provide a proof of the assertion or a counterexample?
>
>Best
>
>Diana
>Professor Diana Kornbrot
> School of Psychology
> University of Hertfordshire
> College Lane, Hatfield, Hertfordshire AL10 9AB, UK
>
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Antony Fielding
Professor of Social and Educational Statistics
Department of Economics
University of Birmingham ,
United Kingdom
Honorary Visiting Fellow
Centre for Multilevel Modelling
Graduate School of Education
University of Bristol
Joint Editor, Journal of the Royal Statistical Society, Series A
(Statistics in Society)
Address: Department of Economics, University of Birmingham, Birmingham B15
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