Comparison  between  Qmax  and  Qmg

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      Symbols:        Significant       Not significant      Total                      (In this site

      Null true                F = a×m₀         m₀ - F                m₀                               a = alpha)     

      Alternative true    T                      m₁ - T                m₁                           

      Total                      S                      m  - S            _   m_ 

      In the case that all experiments with true alternative hypotheses give significant results (p<a), then  Qmg = Qmax  (because  T = m₁ ;  an explanation follows below).

      If  m₁-T = 0,  Qmg = Qmax;    

      if  m₁-T > 0,  Qmg < Qmax.

  Proof:   m = m₀ + m₁ .  In the case that all experiments

with true alternative hypotheses give significant results

(p<a), then   m = m₀+T  and  T= m₁

Qmg = a×m₀ /S ;  S= F+T ;  F =a×m₀      

   (a×m₀+T)/ S =1.     If  m₁-T = 0: 

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

 = [((m₀+T) / S)-(a×m₀+T) / S] / [(1/a)-1] =

 = [(m₀+T -a×m₀ -T) / S] / [(1/a)-1] =

 = [(m₀ -a×m₀) / S] / [(1/a)-(a/a)] =

 = [a×m₀(1-a) / S] / (1-a)  = a×m₀ / S  = Qmg

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        Some examples:

    A)  a = 0.1 ; m₀ = 30,000;  a×m₀ = 3000; 

    (1)  m =   87,000 ;  m₁-T = 0  ;    T = m₁ = 57,000

          S  =   60,000 ;    Qmg = Qmax = 0.05

    (2)  m = 187,000 ;  m₁-T = 100,000 ;    T = 57,000 

          S  =   60,000 ;    Qmg = 0.05 < Qmax = 0. 235

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    B)  a = 0.5 ;  m₀ = 30,000;  a×m₀ =15,000; 

    (3)  m =   87,000 ;  m₁-T = 0 ;    T = m₁ = 57,000

          S  =   72,000 ;     Qmg = Qmax = 0. 208

    (4)  m = 187,000 ;  m₁-T = 100,000 ;    T = 57,000 

                                                                  S  = 172,000 ;     Qmg = Qmax = 0.0872 

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I beg to be notified about any

mistakes that may exist above!

branko.soric@zg.t-com.hr

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