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One-way Analysis of Variance

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Grouped observations

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 )

_ .

_ .

#~v_{~i} = ( 0, ... ,0, _ . . . _ 1, ... ,1, _ . . . _ 0, ... , 0 ) _ _ _ _ [~i^{th} group]

_ .

_ .

#~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}

Testing for uniform mean

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 )