Ms a ss a a 1 ms b ss b b 1 ms a b ss a b a 1 b 1 ms

MS ( A ) =
SS ( A )
a - 1
MS ( B ) =
SS ( B )
b - 1
MS ( A : B ) =
SS ( A : B )
( a - 1)( b - 1)
MS ( err ) =
SS ( err )
ab ( c - 1)

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CHAPTER 11. ANALYSIS OF VARIANCE 211
11.2.1 Two-way ANOVA with Replications
Assume that the number c of replications is the same for all combinations of factor levels and
is greater than 1. Consider the following null hypotheses and their associated test statistics.
H
A : B
:
∑
a
i =1
∑
b
j =1
γ 2
ij
= 0. There are no interactions. F
A : B
=
MS ( A : B )
MS ( err )
H
A
:
∑
a
i =1
α 2
i
= 0. The main effects for factor A are all zero. F
A
=
MS ( A )
MS ( err )
H
B
:
∑
b
j =1
β 2
j
= 0. The main effects for factor B are all zero. F
B
=
MS ( B )
MS ( err )
It follows from 11.2 that if H
A : B
is true
F
A : B
∼ FDist (( a - 1)( b - 1) , ab ( c - 1))
the F distribution with ( a - 1)( b - 1) degrees of freedom in the numerator and ab ( c - 1)
degrees of freedom in the denominator. Similarly, if H
A
is true, F
A
∼ FDist ( a - 1 , ab ( c - 1))
and if H
B
is true, F
B
∼ FDist ( b - 1 , ab ( c - 1)). To test the hypothesis of no interactions
at a given significance level α , reject H
A : B
if F
A : B
> f
α
(( a - 1)( b - 1) , ab ( c - 1)), where f
α
is the 100(1 - α ) percentile of the F distribution. Alternatively, reject H
A : B
if the p-value
is very small.
If H
A : B
is not rejected, we may go on to test the hypothesis of no additive effects for the fac-
tor of primary interest can be tested, for example, by rejecting H
A
if F
A
> f
α
( a - 1 , ab ( c - 1)).
The results of these calculations are traditionally displayed in a 2-way ANOVA table, which
is organized something like the following.
SOURCE SS DF MS F p-val
A SS(A) a - 1 MS(A) F
A
Pr ( F
A
> Obs. )
B SS(B) b - 1 MS(B) F
B
Pr ( F
B
> Obs. )
A:B SS(A:B) ( a - 1)( b - 1) MS(A:B) F
A : B
Pr ( F
A : B
> Obs.)
Error SS(err) ab ( c - 1) MS(err)
Total SS(tot) abc - 1
Example 11.3. The data from a simulation of the split-plot agricultural experiment is
given below in ”wide” format. The experiment is replicated five times for each combination
of plot and variety.

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CHAPTER 11. ANALYSIS OF VARIANCE 212
Plots Varieties Yields.1 Yields.2 Yields.3 Yields.4 Yields.5
1 P1 A 100 87 107 79 95
2 P2 A 99 65 109 90 98
3 P3 A 121 89 84 87 85
4 P4 A 81 77 107 95 103
5 P5 A 77 84 72 86 87
6 P1 B 70 121 112 78 64
7 P2 B 132 97 140 125 143
8 P3 B 108 95 152 108 99
9 P4 B 95 107 120 126 107
10 P5 B 99 82 83 89 92
11 P1 C 65 112 98 95 64
12 P2 C 99 138 111 108 100
13 P3 C 86 133 135 113 105
14 P4 C 104 109 106 85 102
15 P5 C 107 92 105 117 100
16 P1 D 70 93 108 70 99
17 P2 D 112 108 87 66 100
18 P3 D 82 107 113 96 109
19 P4 D 114 95 128 78 104
20 P5 D 83 83 67 117 90
Visual inspection of the data is easy in this format but for R is is better to have only one
observation for each record. The data is available in both the wide format and the ”long”
format. The first few rows of the data in long format are:
> Agexp[1:10,]
Plots Varieties Yields
1 P1 A 100
2 P2 A 99
3 P3 A 121
4 P4 A 81
5 P5 A 77
6 P1 B 70
7 P2 B 132
8 P3 B 108
9 P4 B 95
10 P5 B 99
To produce the ANOVA table with R, we first use the ”lm” function to create a linear model
object and then use the ”anova” function. Notice the use of the operator ”*” in the formula.
It means that both additive terms and interaction terms are to be included.