9.5. Two-Way ANOVA 533Since these are sums of squares, the minimizing values, (mles), must beˆμ=X··,ˆαi=Xi·−X··,andβˆj=X·j−X··. (9.5.7)Note that we have used random variable notation. So these are the maximum
likelihood estimators. It then follows that the maximum likelihood estimator ofσ^2
is
ˆσΩ^2 =∑a
i=1∑b
j=1[Xij−Xi·−X·j+X··]2ab=dfnQ′ 3
ab, (9.5.8)where we have defined the numerator of ˆσΩ^2 as the quadratic formQ′ 3. It follows
from an advanced course in linear models thatabˆσΩ^2 /σ^2 has aχ^2 ((a−1)(b−1))
distribution.
Next we construct the likelihood ratio test forH 0 B. Under the reduced model
(full model constrained byH 0 B),βj=0forallj=1,...,b. To obtain the mles for
the reduced model, the identity (9.5.6) becomes
SS = ab[x··−μ]^2 +b∑ai=1[αi−(xi·−x··)]^2+a∑bj=1[x·j−x··]^2 +∑ai=1∑bj=1[xij−xi·−x·j+x··]^2. (9.5.9)Thus the mles forαiandμremain the same as in the full model and the reduced
model maximum likelihood estimator ofσ^2 isˆσ^2 ω={
a∑b
j=1[X·j−X··](^2) +∑a
i=1
∑b
j=1[Xij−Xi·−X·j+X··]
2
}
ab
. (9.5.10)
Denote the numerator of ˆσ^2 ωbyQ′. Note that it is theresidual variationleft after
fitting the reduced model.
Let Λ denote the likelihood ratio test statistic forH 0 B. Our derivation is similar
to the derivation for the likelihood ratio test statistic for one-way ANOVA of Section
9.2. Hence, similar to equation (9.2.9), our likelihood ratio test statistic simplifies
to
Λab/^2 =
ˆσ^2 Ω
ˆσω^2=
Q′ 3
Q′.Then, similar to the one-way derivation, the likelihood ratio test rejectsH 0 Bfor
large values ofQ′ 4 /Q′ 3 , where in this case,
Q′ 4 =a∑bj=1[x·j−x··]^2. (9.5.11)Note thatQ′ 4 =Q′−Q′ 3 ; i.e., it is the incremental increase in residual variation if
we use the reduced model instead of the full model.
To obtain the null distribution ofQ′ 4 , notice that it is the numerator of the sample
variance of the random variables
√
aX· 1 ,...,√
aX·b. These random variables are