Date: Dec 5, 2012 10:32 AM
Author: Luis A. Afonso
Subject: p-values ranges under Null Hypotheses

p-values ranges under Null Hypotheses

Under Null Hypotheses true p-values do follow a Uniform Distribution when the Test Statistics is continuous.
Given a set n of p-values the probability that the range be < x is easily calculated: for example:
_____ n=8 provides
______x_____P (range<x) __
_____0.500____.0352__
_____0.525____.0475__
_____0.550____.0632__
_____0.575____.0826__
Therefore if we observe a range of 0.525 or less we reject uniformity at 4.75% confidence level, H0 is false; on contrary 0.550 or more there is not sufficient evidence to reject uniformity.
The former case, low range set, if we are dealing with a right side hypotheses test, induces that the values are anomalous near 1: the observed statistical p-values are oddly away from 0, not sufficiently dispersed in [0, 1] which happens if the null hypotheses is really true.

Luis A. Afonso

REM "U-range"
CLS
DEFDBL A-Z
INPUT " sample size "; n
PRINT : PRINT : PRINT : CLS
PRINT : PRINT
PRINT " UNIFORM distribution: Dudewicz, Mishra, p.285 "
PRINT " Modern Mathematical Statistics, Aug. 1987 "
PRINT " prob. Range < x "
REM
DEF fnu (n, x) = n * (n - 1) * (x ^ (n - 1) / (n - 1) - x ^ n / n)
REM
PRINT " x prob. "
FOR x = .5 TO 1.025 STEP .025
COLOR 7
u = fnu(n, x)
IF u < .055 THEN COLOR 14
IF u > .1 THEN GOTO 4
PRINT USING " .### .#### "; x; fnu(n, x)
4 NEXT x
PRINT " sample size = "; n
END