Suppose we have k groups of observations ( of the same variable ) , where the ni observations in each group [&sum._{~i = 1...~k} ~n_{~i} = ~n] are assumed to have the same distribution, i.e. all the observation in a group have the same mean, and the variance is the same for all the observations:
H_1: #~y &tilde. N ( #{&mu.}, &sigma.^2#I ) , _ #{&mu.} = ( &mu._{1,1}, ... , &mu._{1,~n_1}, _ . . . _ &mu._{~k,1}, ... , &mu._{~k,~n_{~k}} ) _ and _ &mu._{~i,~j} = &theta._{~i}. _ So #{&mu.} &in. L_1, dim ( L_1 ) = ~k, and L_1 is spanned by #~v_1, ... , #~v_{~k} where:
#~v_1 = ( 1, ... ,1,0, ... ,0, _ . . . _ 0, ... , 0 )
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_ .
#~v_{~i} = ( 0, ... ,0, _ . . . _ 1, ... ,1, _ . . . _ 0, ... , 0 ) _ _ _ _ [~i^{th} group]
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#~v_{~k} = ( 0, ... ,0, _ . . . _ 0, ... ,0,1, ... ,1 )
Now ( #~y - p_1 ( #~y ) ) &dot. #~v_{~i} = #0, _ ~i = 1, ... ~k _ , _ i.e.
#~y &dot. #~v_{~i} _ = _ p_1 ( #~y ) &dot. #~v_{~i}, _ _ _ _ &forall. ~i
but
p_1 ( #~y ) &dot. #~v_{~i} _ = _ ~n_{~i} est{&theta._~i}
#~y &dot. #~v_{~i} _ = _ sum{~{x_{i,j}},j = 1,~n_{~i}}
so
est{&theta._~i} _ = _ fract{1,~n_{~i}}sum{~{x_{i,j}},j = 1,~n_{~i}} _ = _ ${~y}_{~i} _ _ _ _ _ the group mean
p_1 ( #~y ) _ = _ ( ${~y}_1, ... ${~y}_1, _ _ . . . _ _ ${~y}_{~i}, ... ${~y}_{~i}, _ _ . . . _ _ ${~y}_{~k}, ... ${~y}_{~k} )
RSS _ = _ || #~y - p_1 ( #~y ) || ^2 _ = _ sum{,~i,}sum{ ( ~y_{~i, ~j} - ${~y}_{~i} ) ^2,~j,}
~s_1^2 _ = _ fract{RSS,~n - ~k}
Consider the hypothesis H_2: &theta._1 = &theta._2 = ... = &theta._{~k} = &theta.
This is the same as the hypothesis in " Uniform Normal Distribution ", so the RSS [with n - 1 degrees of freedom] in that model now becomes the TSS in this one:
TSS _ = _ || #~y - p_2 ( #~y ) || ^2 _ = _ sum{ ( ~y_{~i, ~j} - ${~y} ) ^2,~i{,} ~j,} _ _ _ _ _ where ${~y} _ = _ fract{1,~n}rndb{ sum{~y_{~i, ~j}, ~i {,} ~j,}}
~s_2^2 _ = _ fract{1,~n - 1} sum{ ( ~{x_i} - ${~y} ) ^2,1,~n}
Also
ESS _ = _ || p_1 ( #~y ) - p_2 ( #~y ) || ^2 _ = _ sum{~n_{~i} ( ${~y}_{~i} - ${~y} ) ^2,~i = 1,~k}
Anova Table:
| Sum of Squares | df | MS | ~r | |
| ESS = | sum{~n_{~i} ( ${~y}_{~i} - ${~y} ) ^2,~i ,} | ~k &minus. 1 | fract{ESS,~k &minus. 1} | fract{ESS / ~k &minus. 1,RSS / ~n - ~k} |
| RSS = | sum{,~i,}sum{ ( ~y_{~i, ~j} - ${~y}_{~i} ) ^2,~j,} | ~n - ~k | fract{RSS,~n - ~k} = ~s_1^2 | |
| TSS = | sum{ ( ~y_{~i, ~j} - ${~y} ) ^2,~i{,} ~j,} | ~n - 1 | fract{TSS,~n - 1} = ~s_2^2 | |
Now
~r _ = _ fract{ESS / ~k &minus. 1,RSS / ~n - ~k} _ _ ~ _ _ F ( ~k - 1, ~n - ~k )