Der  Fehler in der ANOVA/t,(t²)-Test

[1]      Modell                   Ω    =  X   +   nA_m   | - µΩ ,    | H₁

[2]                                       ω_ij   =  x_ij  + (a_j v α_j) ,                 |H₁       random v fixed 

[3]     Konstrukt         σ²_Ω  =  σ²_x + nσ²_a    ;       random 

[4]                                     σ²_Ω  =  σ²_x + ns²_α   ;        fixed 

[5]                         σ²_zwischen   =  σ²_x + nσ²_a   ;         random

[6]                            σ²_zwischen   =  σ²_x + ns²_α   ;         fixed

[7]                           σ²_innerhalb  =  σ²_x                ;       random , fixed

[8]            E(F)  =  σ²_zwischen / σ²_innerhalb  =  σ²_x + n(σ²_a v s²_α) / σ²_x

[9]      Modell             σ²_Ω      =  σ²_x + σ²_a   ;           random,  

[10]                                σ²_Ω      =  σ²_x + s²_α    ;           fixed,      

[11]   Beweis -Preisfrage       (σ²_Ω - σ²_X) = σ²(nA_m) 

[12]                                                                        =  σ²_a  v  nσ²_a 

[13]   s²(nA_m) =numerisch=  s²( 7,4,1,7,4,1) = s²(7,4,1) = 6 , (ein Exel-Klick genügt)

[14]                          σ²(nA_m) = σ²_a v  s²_α  ,   σ²(nA_m) = σ²_a

[15]        σ²_zwischen  ist nicht modell-konform,  n  ist systematischer Fehler

Lösung I 

[16]             [  (σ²_x + nσ²_a) /σ²_x   =   1 + nσ²_a /σ²_x  ] 

[17]                               E(F_neu)  =(11)= [(E(F) - 1) / n] +1 ;                                                  

[18]                              E(F_neu)     =    (σ²_x + σ²_a) / σ²_x  ;         „F_neu“ approximativ

Lösung II 

[19]             σ²_{zwischen Ω^T} =  σ²_x + σ²_a ,       σ²_innerhalbΩ^T  = σ²_x 

[20]             E[F_Ω^T]^-1   =  (σ²_x + σ²_a) / σ²_x ;      random

Alternativen 

[21]  vorher/nacher     E(F)  =  σ²_{Ω experimentell a} /σ²_{Ω kontroll b} ;   

                                                                                                           a, b := vorher gruppiert

[22]                                                =   σ²_x +σ²_avα / σ²_x  ;      fixed, random 

[22]  partitioniert          E(F) =  σ²_{Ω 1} / σ²_2innerhalb ;     1,2 := nacher randomisiert

[23]                                               =   σ²_x +σ²_avα / σ²_x  ;       fixed, random

allgemeines Problem 

[24]    F_approximativ  =    R ( s²_x ,  s²_avα , 2s_ξδ , 2s_x(avα),  Umfänge /Freiheitsgrade )

P.S.

[25]     Konstrukt     H₀:       µ_.j = µ_.1 = µ_.2 = … = µ_.m ;      σ²(µ_.j) = 0 , heuristisch 

[26]     Dilemma                     0  = nσ²(µ_.j) ≠ σ²_x 

[27]     Modell           H₀:        µ_.j ≈ µ_.1 ≈ µ_.2 ≈ … ≈ µ_.m ;      σ²(µ_.j) ≠ 0 

[28]     Modell                         Ω = X  |-µΩ ,   |H₀                                                                                  

[29]     Modell                       ω_ij  = x_ij ,             |H_o                                                 

[30]     Modell              nσ²(µ_.j) = σ²_x ,            | Normalverteilung

[31]    Das Beschriebene gilt ebenso für den   t,(t²)-Test

Literaturliste

SCHACH, SIEGFRIED  und  SCHÄFER, THOMAS

Regressions- und Varianzanalyse (Hochschlultext)

New York, Springer-Verlag 1978

SCHEFFÈ, HENRY   

The Analysis of  Variance

New York,  Wiley-Interscience 1999

SEARLE, SHAYLE R. ; CASSELLA, GEORGE;  McCULLOCH, CHARLES E

Variance Components    Ithaca, New 1992