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Luis A. Afonso
Posts:
4,725
From:
LIsbon (Portugal)
Registered:
2/16/05


Fisher´s H: k independent evaluations for True Null
Posted:
Aug 9, 2013 5:44 PM


Fisher´s H: k independent evaluations for True Null
___1___Suppose we have k=2 pvalues 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 pvalues 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 pvalues is not, necessarily, a notsignificant twice stated: on contrary 9.4% (0.47/5.00) times leads to a significant result. On the other hand a one significant pvalue joined with a nosignificant 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.p2H 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 _________pvalues _________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 AZ RANDOMIZE TIMER all = 10000000 REM FOR I = 1 TO all LOCATE 5, 40 PRINT USING "##########"; all  I 4 X = RND: Y = RND IF X < 1E10 OR Y < 1E10 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 AZ PRINT " HOW MANY SIGNIFICANT pVALUES 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 < 1E20 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 " ingroup " LOCATE 8, 40: PRINT " signif. " LOCATE 9, 40: PRINT " pvalues Hsignf. Cumulative" sum = 0: ty = 5 FOR t = 0 TO nn: REM t = signficant pvalues 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



