456 Chapter 10:Analysis of Variance
where
∑m
i= 1
αi=
∑n
j= 1
βj= 0
The valueμis called thegrand mean,αiis thedeviation from the grand mean due to row i,
andβjis thedeviation from the grand mean due to column j.
Let us now determine estimators of the parametersμ,αi,βj,i=1,...,m,j=1,...,n.
To do so, continuing our use of “dot” notation, we let
Xi.=
∑n
j= 1
Xij/n=average of the values in rowi
X.j=
∑m
i= 1
Xij/m=average of the values in columnj
X..=
∑m
i= 1
∑n
j= 1
Xij/nm=average of all data values
Now,
E[Xi.]=
∑n
j= 1
E[Xij]/n
=μ+
∑n
j= 1
αi/n+
∑n
j= 1
βj/n
=μ+αi since
∑n
j= 1
βj= 0
Similarly, it follows that
E[X.j]=μ+βj
E[X..]=μ
Because the preceding is equivalent to
E[X..]=μ
E[Xi.−X..]=αi
E[X.j−X..]=βj