Date: Aug 9, 2013 5:44 PM Author: Luis A. Afonso Subject: Fisher´s H: k independent evaluations for True Null 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