Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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: