However, in fact, we don't need Qmax or the calculation below** because we can find Qmg' directly from the histogram!***

      Storey and Tibshirani (as well as others) calculate the q-values (or the FDR, etc.), although it seems that the proportion of false discoveries (Qmg) can directly and simply be found from the histogram(?).  Do q-values give better (smaller) estimates of the true proportion of false discoveries than the Qmg?

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** Qmg =  F/S = F/ (F+T) ;    Qmax = [(m/S)-1] / [(1/a)-1];   a = 0.5 ;  a' = 0,05  

      If m₁-T = 0,  then  Qmg =Qmax  (as explained on the page "Qmg & Qmax").  

      We know the following approximate values (in thousands):

m = 84 ,   S = F+T = 56  (for a = 0.5),   S' = F'+T' = 16.5  (for a' = 0.05) 

      If we take:  a = 0.5 (in the case shown in Fig. 1. above)  and if  m₁-T = 0,  then we find:  Qmg = Qmax = [(m/S)-1] / [(1/a)-1] = 0.5 .     This may be far from the actual proportion of false discoveries, but we can improve the estimate by proceeding to the "second step":

      F/S = Qmax;  F =S×Qmax =56×0.5 =28 ;  F'= F×a'/a  = S×Qmax×a'/a  = 2.8 

Qmax' =  F'/ S' ;  Qmax' (= Qmg')  = (S / S')×Qmax×a'/a  =  2.8/16.5  =  0.1697

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***   Alternatively, we find directly from the histogram (approximately):

m₀ -F = 28 ;   F' = 2×(m₀ - F)×a'= 56×0.05 = 2.8 ; 

       Qmg' = F'/S' = [2×(m₀ - F)×a'] / S' = 2.8/16.5 =  0.1697   (in 16,500 p-values)

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      So, the Qmax method, and especially the improved "two-step Qmax method", can sometimes be as good as the "Qmg graphical method", but the former methods seem to be unnecessary, because they have no advantage over Qmg, which, moreover, is simpler to estimate!

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      Below is a result (FDR) obtained by applying the Benjamini-Hochberg method to the same example (Fig. 1. in the paper by Storey and Tibshirani)!

      We choose, say, alpha = a = 0.05 and apply the Benjamini-Hochberg method:

p₂₃₅₀₀ = 0.1 > (23500/ 84000)×0.05 = 0.014

p₁₆₅₀₀ = 0.05 > (16500/ 84000)×0.05 = 0.0098

      Hence, the set of p-values has to be smaller than 16,500 in order that the null hypotheses can be rejected at the level  a = 0.05 ;   so, perhaps we could reach

FDR = 0.05   -  but only for a much smaller set!

      On the other hand, if we choose alpha = a = 0.255  we have:

p₂₃₅₀₀  = 0.1 > (23500/84000)×0.255= 0.071

p₁₆₅₀₀ = 0.05 < (16500/84000)×0.255= 0.0501

      FDR = 0.255  for the set of 16,500 p-values (approximately); 

      (FDR > Qmg' = Qmax')  

      Is the above calculation correct or wrong? Please, tell me if I have mistakenly applied the Benjamini-Hochberg method!

      I also beg you to answer this question:

      In the above example, can the calculation of q-values, or any other methods for estimating the proportion of false discoveries, give better results than the histogram (i.e. Qmg method); or, perhaps, are all such methods no better than Qmg, at least in very large sets of p-values?

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REFERENCES:

      1. Sorić, B. (1989). Statistical "discoveries" and effect-size estimation. J. Amer. Statist. Assoc., 84, 608-610. http://www.jstor.org/pss/2289950

      2.  Storey JD, Tibshirani R. (2003) Statistical significance for genome-wide studies. Proc Natl Acad Sci 100: 9440-9445. http://genomics.princeton.edu/storeylab/papers/Storey_Tibs_PNAS_2003.pdf 

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