534 Inferences About Normal Linear Modelsindependent with the commonN(√
aμ, σ^2 ) distribution; see Exercise 9.5.2. Hence,
by Theorem 3.6.1,Q′ 4 /σ^2 hasχ^2 (b−1) distribution. In a more advanced course, it
can be further shown thatQ′ 4 andQ′ 3 are independent. Hence, the statistic
FB=a∑b
j=1[X·j−X··](^2) /(b−1)
∑a
i=1
∑b
j=1[Xij−Xi·−X·j+X··]
(^2) /(a−1)(b−1)
(9.5.12)
has anF(b− 1 ,(a−1)(b−1)) underH 0 B. Thus, a levelαtest is to rejectH 0 Bin
favor ofH 1 Bif
FB≥F(α, b− 1 ,(a−1)(b−1)). (9.5.13)
If we are to compute the power function of the test, we need the distribution of
FBwhenH 0 Bis not true. As we have stated above,Q′ 3 /σ^2 , (9.5.8), has a central
χ^2 -distribution with (a−1)(b−1) degrees of freedom under the full model, and,
hence, underH 1 B. Further, it can be shown thatQ′ 4 , (9.5.11), has a noncentralχ^2 -
distribution withb−1 degrees of freedom underH 1 B. To compute the noncentrality
parameters ofQ′ 4 /σ^2 whenH 1 Bis true, we haveE(Xij)=μ+αi+βj,E(Xi.)=
μ+αi,E(X.j)=μ+βj,andE(X..)=μ. Using the general rule discussed in
Section 9.4, we replace the variables inQ′ 4 /σ^2 with their means. Accordingly, the
noncentrality parameterQ′ 4 /σ^2 is
a
σ^2
∑b
j=1
(μ+βj−μ)^2 =
a
σ^2
∑b
j=1
βj^2.
Thus, if the hypothesisH 0 Bis not true,Fhas a noncentralF-distribution withb− 1
and (a−1)(b−1) degrees of freedom and noncentrality parametera
∑b
j=1β
2
j/σ
(^2).
A similar argument can be used to construct the likelihood ratio test statistics
FAto testH 0 AversusH 1 A, (9.5.3). The numerator of theF test statistic is the
sum of squares among rows. The test statistic is
FA=
b
∑a
i=1[Xi·−X··]
(^2) /(a−1)
∑a
i=1
∑b
j=1[Xij−Xi·−X·j+X··]
(^2) /(a−1)(b−1)
(9.5.14)
and it has anF(a− 1 ,(a−1)(b−1)) distribution underH 0 A.
9.5.1 InteractionbetweenFactors...................
The analysis of variance problem that has just been discussed is usually referred
to as atwo-way classification with one observation per cell. Each combination ofi
andjdetermines a cell; thus, there is a total ofabcells in this model. Let us now
investigate another two-way classification problem, but in this case we takec> 1
independent observations per cell.
LetXijk,i=1, 2 ,...,a, j=1, 2 ,...,b,andk=1, 2 ,...,c,denoten=abc
random variables that are independent and have normal distributions with common,
but unknown, varianceσ^2. Denote the mean of eachXijk,k=1, 2 ,...,c,byμij.