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Topic: Fisher´s H: k independent evaluations for True Null
Replies: 2   Last Post: Aug 13, 2013 5:48 AM

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Luis A. Afonso

Posts: 4,518
From: LIsbon (Portugal)
Registered: 2/16/05
Fisher´s H: k independent evaluations for True Null
Posted: Aug 9, 2013 5:44 PM
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Fisher´s H: k independent evaluations for True Null

___1___Suppose we have k=2 p-values concerning the same Null Hypotheses Significance Test: Based on the property that they follows an Uniform [0, 1] Distribution if H0 is true one can find out the duo p-values frequencies leading to a Fisher´s H at 5% significance and from total (program <TOFIND>):
_______________________________from Signf.____from total__
a)_p1, p2 both significant(5%)_______ __________________________________0.25%________0.25%____
b)_only one_______________________4.28________4.75_____
c)_any one significant____________0.47_________90.25_____

It´s evident that sometimes the natural, Deductive Logic, could be inappropriate: in fact the correct decision does depend exclusively on the H value (Inductive Logic). For example two not significant p-values is not, necessarily, a not-significant twice stated: on contrary 9.4% (0.47/5.00) times leads to a significant result. On the other hand a one significant p-value joined with a no-significant could not be an inconclusive *tie*: 85.6 % from all significant values indicates a significant H.

_______As a complementary analysis is to find what pairs p1 and p2 leads to a 5% significant Fisher´s H. One gets significance for the result as long as:
_________-2*Log (p1*p2) > H_______H=9.48773
_________2*Log (p1*p2) < -H
_________p2 < Exp(-H/2) / p1 = 0.008704/ p1

Table of maximum p2 under p1

__p1_________max.p2|H significant
_0.05_____________0.174___
_0.10_____________0.087___
_0.20_____________0.044___
_0.30_____________0.029___
_0.40_____________0.022___
_0.50_____________0.017___
_0.60_____________0.015___
_0.70_____________0.012___
_0.80_____________0.011___
_0.90_____________0.010___
_0.95_____________0.009___


___2____k=10, 15, 20

_k=10____signif._____Hcrit=31.41
_________p-values
_________number____Hsignif.____Cumulative__
___________8________0.001______0.001__
___________9________0.011______0.012__
__________10________0.038______0.050__
_k=15_______________43.77
__________13________0.003______0.004__
__________14________0.015______0.018__
__________15________0.031______0.050__
_k=20_______________55.76
__________17________0.001______0.001__
__________18________0.005______0.007__
__________19________0.018______0.024__
__________20________0.026______0.050__


Luis A. Afonso


REM "TOFIND"
CLS
DEFDBL A-Z
RANDOMIZE TIMER
all = 10000000
REM
FOR I = 1 TO all
LOCATE 5, 40
PRINT USING "##########"; all - I
4 X = RND: Y = RND
IF X < 1E-10 OR Y < 1E-10 THEN GOTO 4
g = 9.488
h = -2 * LOG(X) - 2 * LOG(Y)
IF h < g THEN GOTO 100
IF X < .05 AND Y < .05 THEN yesboth = yesboth + 1 / all
IF X < .05 AND Y > .05 THEN nomatter = nomatter + 1 / all
IF X > .05 AND Y < .05 THEN nomatter = nomatter + 1 / all
IF X > .05 AND Y > .05 THEN annie = annie + 1 / all
100 NEXT I
LOCATE 10, 50: PRINT " percent 5% significant "
LOCATE 11, 50: PRINT USING "##.### BOTH signf "; yesboth * 100
LOCATE 12, 50: PRINT USING "##.### ONE "; nomatter * 100
LOCATE 13, 50: PRINT USING "##.### ANY "; annie * 100
END
REM "H1H
CLS
DEFDBL A-Z
PRINT " HOW MANY SIGNIFICANT p-VALUES are enough to get";
PRINT " A SIGNIFICANT FISHERïs H ? "
INPUT " __________k = "; nn
INPUT " __________crit = "; crit
INPUT " __________all = "; all
DIM H(nn)
RANDOMIZE TIMER
FOR i = 1 TO all
LOCATE 5, 40: PRINT USING "##########"; all - i
p = 1: how = 0
FOR ki = 1 TO nn
5 v = RND
IF v < 1E-20 THEN GOTO 5
IF v < .95 THEN how = how + 1: REM how= number signf. values
p = p * v
NEXT ki
REM
H = -2 * LOG(p)
REM
IF H > crit THEN H(how) = H(how) + 1 / all
IF H > crit THEN ncrit = ncrit + 1
NEXT i
REM
COLOR 14
LOCATE 7, 40: PRINT " in-group "
LOCATE 8, 40: PRINT " signif. "
LOCATE 9, 40: PRINT " p-values Hsignf. Cumulative"
sum = 0: ty = 5
FOR t = 0 TO nn: REM t = signficant p-values
sum = sum + H(t)
IF H(t) = 0 THEN GOTO 8
LOCATE 5 + ty, 40
ty = ty + 1
PRINT USING " ## #.### #.### "; t; H(t); sum
8 NEXT t
END



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