Thanks to all the very helpful people who confirmed and supplied arguments
and/or references
YES it IS true
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.
My intuitions were wrong & I feel an idiot
Several supplied intuitiveč arguments, e.g.
Think this way: a probability p means that in any n events p*n will have the
statistic one side of the value and (1-p)*n will be the other side, and if p
were not uniformly distributed some p's would be more likely and hence more
events would occur that side of any other probability.
What percentage of the time (if you imagine repeating the experiment) would
you expect to get a p-value less than 5%? Answer, 5% of the time, that's
what it means to say the p-value is 5%. What percentage of the time would
you get a p-value less than 10%? Answer, 10% of the time. What percentage of
the time would you get a p-value less than x%? Answer, x% of the time. That
is, the probability (as a percentage) of getting a p-value less than x% is
x%.
So if y is a number between 0 and 1, what is the probability that the
p-value will be less than y? Answer: the probability is y. This implies that
the p-value has a uniform (rectangular) distribution on (0,1).
References include:
Sackrowitz and Samuel-Cahn, American Statistician, 1999, 53(4), 326-331
for the distribution of the p value under H0.
Senn, S. (2003). P-Values. Encyclopedia of Biopharmaceutical Statistics, 685
- 695.
Additional useful inofrmation
1. the test statistic must be continuous
2. proofs are by standardč transform methods
3. the facti s much used in simulation and random number generation
Thanks again to all who replied so helpfully and promptly
Best
Diana
______________________________________________
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
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