9.5. Two-Way ANOVA 535Under the additive model, (9.5.1), the mean of each cell depended on its row and
column, but often the mean is cell-specific. To allow this, consider the parameters
γij = μij−{μ+(μi·−μ)+(μ·j−μ)}
= μij−μi·−μ·j+μ,
fori=1,...a,j=1,...,b. Henceγijreflects the specific contribution to the cell
mean over and above the additive model. These parameters are calledinteraction
parameters. Using the second form (9.5.2), we can write the cell means as
μij=μ+αi+βj+γij, (9.5.15)where
∑a
i=1αi=0,∑b
j=1βj=0,and∑a
i=1γij=∑b
j=1γij=0. Thismodelis
called atwo-waymodel with interaction.
For example, takea=2,b=3,μ=5,α 1 =1,α 2 =−1,β 1 =1,β 2 =0,
β 3 =−1,γ 11 =1,γ 12 =1,γ 13 =−2,γ 21 =−1,γ 22 =−1, andγ 23 = 2. Then the
cell means are
Factor B
123
Factor A 1 μ 11 =8 μ 12 =7 μ 13 =3
2 μ 21 =4 μ 22 =3 μ 23 =5
If eachγij= 0, then the cell means are
Factor B
123
Factor A 1 μ 11 =7 μ 12 =6 μ 13 =5
2 μ 21 =5 μ 22 =4 μ 23 =3
Note that the mean profile plots for this second example are parallel, but those in
the first example (where interaction is present) are not.
The derivation of the mles under the full model, (9.5.15), is quite similar to the
derivation for the additive model. LettingSSdenote the sums of squares in the
exponent ofein the likelihood function, we obtain the following identity by adding
in and subtracting out (we have omitted subscripts on the sums):SS =∑∑∑
(xijk−μ−αi−βj−γijk)^2=∑∑∑
{[xijk−xij·]−[μ−x···]−[αi−(xi··−x···)]−[βj−(x·j·−x···)]−[γij−(xij·−xi··−x·j·+x···]}^2
=∑∑∑
[xijk−xij·]^2 +abc[μ−x···]^2 +bc∑
[αi−(xi··−x···)]^2 +ac∑
[βj−(x·j·−x···)]^2 +c∑∑
[γij−(xij·−xi··−x·j·+x···)]^2 (9.5.16)where, as in the additive model, the cross product terms in the expansion are 0.
Thus, the mles ofμ,αiandβjare the same as in the additive model; the mle of
γijis ˆγij=Xij·−Xi··−X·j·+X···;andthemleofσ^2 is
ˆσΩ^2 =∑∑∑
[Xijk−Xij·]^2
abc. (9.5.17)