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