Introduction to Probability and Statistics for Engineers and Scientists

10.4Two-Factor Analysis of Variance: Introduction and Parameter Estimation 455

However, if we letμdenote the average value of theμi— that is,

μ=

∑m

i= 1

μi/m

then we can rewrite the model as

E[Xij]=μ+αi

whereαi =μi−μ. With this definition ofαias the deviation ofμifrom the average
mean value, it is easy to see that
∑m

i= 1

αi= 0

Atwo-factoradditivemodelcanalsobeexpressedintermsofrowandcolumndeviations.
If we letμij=E[Xij], then the additive model supposes that for some constantsai,i=
1,...,mandbj,j=1,...,n

μij=ai+bj

Continuing our use of the “dot” (oraveraging) notation, we let

μi.=

∑n

j= 1

μij/n, μ.j=

∑m

i= 1

μij/m, μ..=

∑m

i= 1

∑n

j= 1

μij/nm

Also, we let

a.=

∑m

i= 1

ai/m, b.=

∑n

j= 1

bj/n

Note that

μi.=

∑n

j= 1

(ai+bj)/n=ai+b.

Similarly,

μ.j=a.+bj, μ..=a.+b.

If we now set

μ=μ..=a.+b.

αi=μi.−μ=ai−a.

βj=μ.j−μ=bj−b.

then the model can be written as

μij=E[Xij]=μ+αi+βj