9.5. Two-Way ANOVA 5319.4.6.Extend the Bonferroni procedure to simultaneous testing. That is, suppose
we havemhypotheses of interest: H 0 iversusH 1 i,i=1,...,m.FortestingH 0 i
versusH 1 i,letCi,α be a critical region of sizeαand assumeH 0 iis rejected if
Xi∈Ci,α, for a sampleXi. Determine a rule so that we can simultaneously test
thesemhypotheses with a Type I error rate less than or equal toα.
9.5 Two-WayANOVA
Recall the one-way analysis of variance (ANOVA) problem considered in Section
9.2 which was concerned with one factor atblevels. In this section, we are con-
cerned with the situation where we have two factorsAandBwith levelsaand
b, respectively. This is called atwo-way analysis of variance (ANOVA). Let
Xij,i=1, 2 ,...,aandj=1, 2 ,...,b, denote the response for factorAat level
iand factor B at levelj. Denote the total sample size byn=ab. We shall assume
that theXijs are independent normally distributed random variables with common
varianceσ^2. Denote the mean ofXijbyμij.Themeanμijis often referred to as
the mean of the (i, j)th cell. For our first model, we consider theadditive model
where
μij=μ+(μi·−μ)+(μ·j−μ) ; (9.5.1)
that is, the mean in the (i, j)th cell is due to additive effects of the levels,iof factor
Aandjof factorB, over the average (constant)μ.Letαi=μi·−μ,i=1,...,a;
βj=μ·j−μ,j=1,...,b;andμ=μ. Then the model can be written more simply
as
μij=μ+αi+βj, (9.5.2)
where∑a
i=1αi=0and∑b
j=1βj= 0. We refer to this model as being atwo-way
additive ANOVA model.
For example, takea=2,b=3,μ=5,α 1 =1,α 2 =−1,β 1 =1,β 2 =0,and
β 3 =−1. 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 =3Note that for eachi, the plots ofμijversusjare parallel. This is true for additive
models in general; see Exercise 9.5.9. We call these plotsmean profile plots.
Had we takenβ 1 =β 2 =β 3 = 0, then the cell means would be
Factor B
123
Factor A 1 μ 11 =6 μ 12 =6 μ 13 =6
2 μ 21 =4 μ 22 =4 μ 23 =4The hypotheses of interest areH 0 A:α 1 =···=αa=0versusH 1 A:αi =0,forsomei, (9.5.3)