Introduction to Probability and Statistics for Engineers and Scientists

10.6Two-Way Analysis of Variance with Interaction 463

10.6 Two-Way Analysis of Variance with Interaction

In Sections 10.4 and 10.5, we considered experiments in which the distribution of the
observed data depended on two factors — which we called the row and column factor.
Specifically, we supposed that the mean value ofXij, the data value in rowiand column
j, can be expressed as the sum of two terms — one depending on the row of the element
and one on the column. That is, we supposed that

Xij∼N(μ+αi+βj,σ^2 ), i=1,...,m, j=1,...,n

However, one weakness of this model is that in supposing that the row and column effects
are additive, it does not allow for the possibility of a row and column interaction.
For instance, consider an experiment designed to compare the mean number of defective
items produced by four different workers when using three different machines. In analyzing
the resulting data, we might suppose that the incremental number of defects that resulted
from using a given machine was the same for each of the workers. However, it is certainly
possible that a machine could interact in a different manner with different workers. That
is, there could be a worker–machine interaction that the additive model does not allow for.
To allow for the possibility of a row and column interaction, let

μij=E[Xij]

and define the quantitiesμ,αi,βj,γij,i=1,...,m,j=1,...,nas follows:

μ=μ..

αi=μi.−μ..

βj=μ.j−μ..

γij=μij−μi.−μ.j+μ..

It is immediately apparent that

μij=μ+αi+βj+γij

and it is easy to check that

∑m

i= 1

αi=

∑n

j= 1

βj=

∑m

i= 1

γij=

∑n

j= 1

γij= 0

The parameterμis the average of allnmmean values; it is called thegrand mean.The
parameterαiis the amount by which the average of the mean values of the variables in