Introduction to Probability and Statistics for Engineers and Scientists

464 Chapter 10:Analysis of Variance

rowiexceeds the grand mean; it is called theeffect of row i. The parameterβjis the amount
by which the average of the mean values of the variables in columnjexceeds the grand
mean; it is called theeffect of column j. The parameterγij=μij−(μ+αi+βj)isthe
amount by whichμijexceeds the sum of the grand mean and the increments due to row
iand to columnj; it is thus a measure of the departure from row and column additivity
of the mean valueμij, and is called theinteraction of row i and column j.
As we shall see, in order to be able to test the hypothesis that there are no row and column
interactions — that is, that allγij=0—itisnecessary to have more than one observation
for each pair of factors. So let us suppose that we havelobservations for each row and
column. That is, suppose that the data are{Xijk,i=1,...,m,j=1,...,n,k=1,...,l},
whereXijkis thekth observation in rowiand columnj. Because all observations are
assumed to be independent normal random variables with a common varianceσ^2 , the
model is

Xijk∼N(μ+αi+βj+γij,σ^2 )

where

∑m

i= 1

αi=

∑n

j= 1

βj=

∑m

i= 1

γij=

∑n

j= 1

γij= 0 (10.6.1)

We will be interested in estimating the preceding parameters, and in testing the following
null hypotheses:

H 0 r:αi=0, for alli

H 0 c:βj=0, for allj

H 0 int:γij=0, for alli,j

That is,H 0 ris the hypothesis of no row effect;H 0 cis the hypothesis of no column effect;
andH 0 intis the hypothesis of no row and column interaction.
To estimate the parameters, note that it is easily verified from Equation 10.8 and the
identity

E[Xijk]=μij=μ+αi+βj+γij

that

E[Xij.]=μij=μ+αi+βj+γij

E[Xi..]=μ+αi

E[X.j.]=μ+βj

E[X...]=μ