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a0 =0.01 ,
a =0.05 , m =50,000
---------------------------------------------------------------------------------------------------------------------------------------- . .
m0 m1
f S Q --> E ≥ Emin <-- Qmax
----------------------------------------------------------------------------------------------------------------
50,000
0 1 500
1 1
1 1
50,000
0 0,5
500 1
1 1 1
50,000
0 0.05
500 1
1
1 1
50,000
0 0.01
500 1
1
1 1
50,000 0
near 0 500 1 1 1 1
---------------------------------------------------------------------------------------------
38,000
12,000 1 12,380 0.03069 0.07916 0.07916 0.03069
38,000 12,000
0.9 11,180 0.03399 0.08766
0.08332
0.03507
38,000 12,000
0.8 9980 0.03808 0.09820 0.08848 0.04051
38,000
12,000 0.75
9380 0-04051 0.10448 0.09155 0.04374
38,000
12,000 0.5
6380 0.05956 0.15361 0.11561 0.06906
38,000
12,000 0.25
3380 0.11243 0.28994 0.18235 0.13932
38,000
12,000 0.05
980 0.38776 1 0.53000 0.50526
38,000
12,000 0.01
500 0.76000 1
1 1
38,000 12,000 near 0 380 1
1 1 (1)
----------------------------------------------------------------------------------------------------------------
25,000 25,000 1 25,250 0.00990 0.05941 0.05941 0.0099
25,000 25,000
0.9 22,750 0.01099
0.06593 0.06149 0.01210
25,000 25,000 0.75 19,000
0.01316 0.07895 0.06566 0.01648
25,000 25,000 0.5 12,750 0.01961 0.11765 0.07803 0.02951
25,000 25,000 0.25 6500 0.03846 0.23077 0.11422 0.06760
25,000 25,000 0.05 5100 0.16667 1 0.36027 0.32660
25,000
25,000 0.01
500 0.5 (1) 1 1
25,000 25,000 near 0 250
1
1
1 (1)
----------------------------------------------------------------------------------------------------------------
12,000
38,000 1
38,120 0.00315 0.05299 0.05299 0.00315
12,000
38,000 0.9
34,320 0.00350 0.05886 0.05439 0.00462
12,000 38,000 0.75 28,620
0.00419 0.07058 0.05717 0.00755
12,000
38,000 0.5
19,120 0.00628 0.10565 0.06549 0.01631
12,000
38,000 0.25
9620 0.01247 0.20998 0.09028 0.04240
12,000 38,000 0.05 2020
0.05941 1 0.27792 0.23992
12,000
38,000 0.01
500 0.24
(1) 1
1
12,000 38,000
near
0 120
1 1 1 1
----------------------------------------------------------------------------------------------------------------
0 50,000 1
50,000 0
0.05 0.05 0
0 50,000
0.9
45,000 0
0.05556
0.05106
0.00112
0 50,000
0.75
37,500 0 0.06667 0.05320 0.00337
0 50,000
0.5
25,000 0 0.1
0.05959 0.01010
0 50,000
0.25 12,500 0
0.2
0.07878 0.03030
0 50,000
0.05
2500 0 1 0.23232 0.19192
0 50,000 0.01 500
0 (1) 1 1
0
50,000 0.000001 0.05 0 (1) 1
1
---------------------------------------------------------------------------------------------
REMARKS:
---
1. If the true proportion (Q) of false discoveries can be
estimated rather accurately in a very large set (either by means of Qmg or some other method),
then this estimate of Q can be used to calculate f and E (=proportion of false confidence intervals). In the table above it is shown how the values m (=m0+m1),
f , S , Q , Qmax and E depend
on each other. If we use Qmax instead of Q for calculating E, then we obtain
the values that are denoted as Emin ; (Emin ≤ E).
...
2.
Qmg might be used
to estimate the proportion (E) of false confidence intervals in a large set S. If we want to take into account random variations, we can
calculate the practically "maximal possible" and "minimal possible" values of Qmg. Namely,
the observed proportion Qmg can randomly differ from the unknown actual probability (p=F/S) of making a false discovery in
the set of declared discoveries.
Suppose, for
example, that a0 = 0.01 , a = 0.05 , m = 50,000 , S=19,000
and we find: Qmg (=F/S) = 250/19,000 =0.01316 . If Qmg ≈ Q, the resulting values are:
f = 0.75 and E = 0.07895 i.e. near 8 percent.
If we suppose that the actual probability is p=0.01737 = 330/19,000
, we can calculate the probability (p*) of obtaining 250 false discoveries (or fewer) in 19,000 declared discoveries with
p=0.01737 which is p*= 0.0000044 . Likewise,
we may suppose that the actual probability is p=190/19,000 =0.01, and we shall find that the probability (p**) of obtaining
250 false discoveries (or more) in 19,000 declared discoveries with p=0.01 is p**= 0.0000061 .
So, we are almost sure that the actual Qmg is between 0.01 and 0.01737 (i.e.: 0.01 < Qmg < 0.017), and the resulting value of E
is (approximately) between 7 percent and 9 percent
(0.067 <
E < 0.092 ; 0.607 < f < 1).
---------------------------------------------
I beg to be notified about any
mistakes that may exist above!
branko.soric@zg.t-com.hr
Go to: Home , E & Emin
-----------------------------------------------------
June - September, 2009
Branko Soric
................................................................................................................................................................
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