536 Inferences About Normal Linear Models
LetQ′′ 3 denote the numerator of ˆσ^2.
The major hypotheses of interest for the interaction model are
H 0 AB: γij=0foralli, jversusH 1 AB:γij =0,forsomei, j. (9.5.18)
Substitutingγij=0inSS, it is clear that the reduced model mle ofσ^2 is
ˆσω^2 =
∑∑∑
[Xijk−Xij·]^2 +c
∑∑
[Xij·−Xi··−X·j·+X···]^2
abc
. (9.5.19)
LetQ′′denote the numerator of ˆσω^2 and letQ′′ 4 =Q′′−Q′′ 3. Then it follows as in the
additive model that the likelihood ratio test statistic rejectsH 0 ABfor large values
ofQ′′ 4 /Q′′ 3. In a more advanced class, it is shown that the standardized test statistic
FAB=
Q′′ 4 /[(a−1)(b−1)]
Q′′ 3 /[ab(c−1)]
(9.5.20)
has underH 0 AB anF-distribution with (a−1)(b−1) andab(c−1) degrees of
freedom.
IfH 0 AB: γij= 0 is accepted, then one usually continues to testαi=0,i=
1 , 2 ,...,a, by using the test statistic
F=
bc
∑a
i=1
(Xi··−X···)^2 /(a−1)
∑a
i=1
∑b
j=1
∑c
k=1
(Xijk−Xij·)^2 /[ab(c−1)]
,
which has a nullF-distribution witha−1andab(c−1) degrees of freedom. Similarly,
the test ofβj=0,j=1, 2 ,...,b, proceeds by using the test statistic
F=
ac
∑b
j=1
(X·j·−X···)^2 /(b−1)
∑a
i=1
∑b
j=1
∑c
k=1
(Xijk−Xij·)^2 /[ab(c−1)]
,
which has a nullF-distribution withb−1andab(c−1) degrees of freedom.
We conclude this section with an example that serves as an illustration of two-
way ANOVA along with its associated R code.
Example 9.5.1. Devore (2012), page 435, presents a study concerning the effects
to the thermal conductivity of an asphalt mix due to two factors: Binder Grade
at three different levels (PG58, PG64, and PG70) and Coarseness of Aggregate
Content at three levels (38%, 41%, and 44%). Hence, there are 3×3 = 9 different
treatments. The responses are the thermal conductivities of the mixes of asphalt at
these crossed levels. Two replications were performed at each treatment. The data
are: