Introduction to Probability and Statistics for Engineers and Scientists

466 Chapter 10:Analysis of Variance

Suppose now that we want to test the hypothesis that there are no row and column
interactions — that is, we want to test

H 0 int:γij=0, i=1,...,m, j=1,...,n

Now, ifH 0 intis true, then the random variablesXij. will be normal with mean

E[Xij.]=μ+αi+βj

Also, since each of these terms is the average oflnormal random variables having variance
σ^2 , it follows that

Var(Xij.)=σ^2 /l

Hence, under the assumption of no interactions,

∑n

j= 1

∑m

i= 1

l(Xij.−μ−αi−βj)^2

σ^2

is a chi-square random variable withnmdegrees of freedom. Since a total of 1+m−
1 +n− 1 =n+m−1 of the parametersμ,αi,i=1,...,m,βj,j=1,...,n, must be
estimated, it follows that if we let

SSint=

∑n

j= 1

∑m

i= 1

l(Xij.−ˆμ−ˆαi−βˆj)^2 =

∑n

j= 1

∑m

i= 1

l(Xij.−Xi..−X.j.+X...)^2

then, underH 0 int,

SSint

σ^2

is chi-square with (n−1)(m−1) degrees of freedom

Therefore, under the assumption of no interactions,

SSint

(n−1)(m−1)

is an unbiased estimator ofσ^2

Because it can be shown that, under the assumption of no interactions,SSeandSSintare
independent, it follows that whenH 0 intis true

Fint=

SSint/(n−1)(m−1)

SSe/nm(l−1)

is anF-random variable with (n−1)(m−1) numerator andnm(l−1) denominator
degrees of freedom. This gives rise to the following significance levelαtest of

H 0 int:allγij= 0