Q 2009

Possible Qmax

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"Maximally possible" value of Qmax

 

      Suppose that we make 5000 experiments with a chosen level of significance a0 = 0.01, and we are ready to tolerate 5 percent false discoveries.  We obtain, say, 2000 discoveries in which  p< 0.01 = a0     (Note:  a  is used instead of alpha).

 

      m = 5000 ;   S = 2000

      Qmax = [(m/S) -1] / [(1/a) -1]  = 0.0152

      If all the null hypotheses were true, we would have  m = m0 = 5000  and we would expect:  0.015000 = 50 false discoveries = F.   However, randomly, F can be larger, but practically not larger than  F=93  (because the probability to obtain more than 93 false discoveries from 5000 experiments with true null hypotheses and with a = 0.01 is about  0.000000001 =10^-9).

      So,  S-F = 2000 -93 = 1907 true discoveries, which means that at the beginning we must have had at the most  5000 - 1907 = 3093 true null hypotheses (instead of 5000 as we had wrongly supposed).  So, we now take m0 =3093 and we expect 0.013093 = 30.93 or 31 false discoveries. Randomly we can obtain more, but practically not more than 65 (because the probability to obtain more is again 0.000000001).   So, F < 65.

      Now we know that in fact  S-F = 2000 -65 = 1935 true alternative hypotheses; 

5000 - 1935 = 3065 true null hypotheses = m0   

      Now we expect 0.013065 = 30.65 , but practically there cannot be more than 64 false discoveries.  Hence, the maximal true proportion of false discoveries is 64/2000 = 0.032  which is more than the expected 0.0152, but this is still tolerable, because it is less than 5 percent.  -  Here are the above calculations once again:

 

  m = 5000                                                             ___       S= 2000          _ 

  m0= 5000                           50000,01 = 50    ---> 93     2000-65 = 1907

  m0= 5000 - 1907 = 3093   30930.01 = 30.9 ---> 65     2000-65 = 1935

  m0= 5000 - 1935 = 3065   30650.01 = 30.7 ---> 64     2000-64 = 1936

  m0= 5000 - 1936 = 3064   30640.01 = 30.6 ---> 64     2000-64 = 1936    

  64/2000 = 0.0320 > 0.0152

 

      If, say,  m = 20.000  or 50,000  and  S = 8000 or 20,000, respectively,  the true proportion of false discoveries would be considerably closer to  0.0152 = Qmax.   

      For example:

 

_m = 20,000                                                                       S= 8000           _ 

  m0= 20,000                         20,0000,01= 200--> 285    8000-285= 7715

  m0= 20,000-7715= 12,285  12,2850.01=123--> 190    8000-190= 7810

  m0= 20,000-7810= 12,190  12,1900.01=122--> 189    8000-189= 7811  

  m0= 20,000-7811= 12,189  12,1890.01=122--> 189    8000-189= 7811 

  189/8000= 0.0236 > 0.0152  

 

 

 

 

 

 

 

 

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