```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 = 10000000REM             FOR I = 1 TO all        LOCATE 5, 40        PRINT USING "##########"; all - I4       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 / all100     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 nn5       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 kiREM        H = -2 * LOG(p)REM        IF H > crit THEN H(how) = H(how) + 1 / all        IF H > crit THEN ncrit = ncrit + 1        NEXT iREM        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 + 1PRINT USING "    ##       #.###       #.###  "; t; H(t); sum8       NEXT t        END
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