All Pairwise Comparisons among Levels of Fertilizer
Fertilizer = 1
subtracted from:
Difference
SE of
Adjusted
Fertilizer
of Means
Difference
T-Value
P-Value
2
0.4000
0.3648
1.097
0.7003
3
1.4750
0.3648
4.044
0.0127
4
2.7250
0.3648
7.471
0.0002
Fertilizer = 2
subtracted from:
Difference
SE of
Adjusted
Fertilizer
of Means
Difference
T-Value
P-Value
3
1.075
0.3648
2.947
0.0650
4
2.325
0.3648
6.374
0.0006
Fertilizer = 3
subtracted from:
Difference
SE of
Adjusted
Fertilizer
of Means
Difference
T-Value
P-Value
4
1.250
0.3648
3.427
0.0316
13-66

Topic 11a, part 4
Two-Way Analysis of
Variance

Objectives
Objectives
Objectives
Objectives
1.
Analyze a two-way ANOVA design
2.
Draw interaction plots
3.
Perform the Tukey test
13-68

Objective 1
Objective 1
Objective 1
Objective 1
• Analyze a Two-Way Analysis of Variance Design
13-69

Recall, there are two ways to deal with factors:
(1) control the factors by fixing them at a single level or by
fixing them at different levels, and
(2) randomize so that their effect on the response variable
is minimized.
In both the completely randomized design and the
randomized complete block design, we manipulated one
factor to see how varying it affected the response variable.
13-70

In a Two-Way Analysis of Variance design, two
factors are used to explain the variability in the
response variable.
We deal with the two factors
by fixing them at different levels.
We refer to the
two factors as factor A and factor B.
If factor A
has
n
levels and factor B has
m
levels, we refer to
the design as an
factorial design.
n
×
m
13-71

Parallel Example 3:
A 2 x 4 Factorial Design
Suppose the rice farmer is interested in comparing the
fruiting period for not only the four fertilizer types, but for two
different seed types as well.
The farmer divides his plot into
16 identical subplots.
He randomly assigns each
seed/fertilizer combination to two of the subplots and
obtains the fruiting periods shown on the following slide.
Identify the main effects.
What does it mean to say there is
an interaction effect between the two factors?
13-72

13-73
Fertilizer 1
Fertilizer 2
Fertilizer 3
Fertilizer 4
Seed
Type A
13.5
13.9
13.5
14.1
15.2
14.7
17.1
16.4
Seed
Type B
14.4
15.0
14.7
15.4
15.3
15.9
16.9
17.3

Solution
The two factors are A:
fertilizer type and B:
seed type.
Since all levels of factor A are combined with all levels of
factor B, we say that the factors are
crossed
.
The main effect of factor A is the change in fruiting period
that results from changing the fertilizer type.
The main effect of factor B is the change in fruiting period
that results from changing the seed type.
We say that there is an interaction effect if the effect of
fertilizer on fruiting period varies with seed type.
13-74

1.
The populations from which the samples are
drawn must be normal.
2.
The samples are independent.
3.
The populations all have the same variance.
Requirements to Perform the Two-Way
Analysis of Variance

In a two-way ANOVA, we test three separate hypotheses.