Chapter 12
UNDERSTANDING THE TWO-WAY
ANALYSIS OF VARIANCE
Going Forward
Your goals in this chapter are to learn:
• What a two-way ANOVA is
• How to calculate main effect means and cell
means
• What a significant main effect indicates
• What a significant interaction indicates
Going Forward
• How to perform the Tukey HSD test on the
interaction
• How to interpret the results of a two-way
experiment
Understanding the Two-Way Design
The Two-Way ANOVA
1.The two-way ANOVA is the parametric
inferential procedure performed when a
design involves two independent variables
2.When both factors involve independent
samples, we perform the two-way betweensubjects ANOVA
The Two-Way ANOVA
3. When both factors involve related measures,
we perform the two-way within-subjects
ANOVA
4. When one factor is tested using independent
samples and the other factor involves
related samples, we perform the two-way
mixed-design ANOVA
Organization
1. Each column represents a level of factor A
2. Each row represents a level of factor B
3. Each square represents combining a level of
factor A with a level of factor B and is called a
cell
4. When we combine all levels of one factor
with all levels of the other factor, the design
is called a factorial design
Organization
Understanding the Main Effects
Main Effects
• The main effect of a factor is the overall effect
changing the levels of that factor has on
dependent scores while we ignore the other
factor in the study.
• To compute a main effect of one factor, we
collapse across the other factor. Collapsing a
factor refers to averaging together all scores
from all levels of that factor.
Main Effects Means
The mean of the level of one factor after
collapsing the other factor is known as the main
effect mean.
Statistical Hypotheses
• Factor A
H 0 :  A1   A2   A3
Ha: At least two of the main effect means are
different
• Factor B
H 0 :  B1   B2
Ha: At least two of the main effect means are
different
Understanding the Interaction Effect
Interaction Effects
The interaction of two factors is called a twoway interaction
 The two-way interaction effect is when the
relationship between one factor and the dependent
variable changes as the levels of the other factor
change
 When you look for the interaction effect, you
compare the cell means. A cell mean is the mean
of the scores from one cell in a two-way design.
Interaction Effect
• An interaction effect is present when the
relationship between one factor and the
dependent scores depends on the level of the
other factor present
• A two-way interaction effect is not present if
the cell means form the same pattern
regardless of the level of the other factor
present
Completing the Two-Way Design
Summary Table
Examining Main Effects
• Each main effect is approached as a separate
one-way ANOVA
• A significant Fobt indicates we should
 conduct a Tukey’s HSD test,
 compute the effect size, and
 graph the means
Examining the Interaction
When the interaction effect is significant, we
need to
• Calculate the effect size using h2,
• Graph the interaction,
• Conduct a Tukey’s HSD test using only
unconfounded comparisons.
Examining the Interaction
• An unconfounded comparison is one in which
two cells differ along only one factor
• When two cells differ along both factors, we
have a confounded comparison
Graphing the Interaction
When graphing the interaction effect,
• Label the means of the dependent variable on
the Y axis,
• Label the factor with the most levels on the X
axis, and
• Plot the cell means of the factor with fewer
levels
• A separate line is plotted for each level
Graphing the Interaction
Interpreting the
Two-Way Experiment
Significant Interaction
• Conclusions about main effects may be
contradicted by the interaction
• The primary interpretation of a two-way
ANOVA may focus on the significant
interaction
• If the interaction is significant, we do not
make conclusions about the main results of
the main effects
Nonsignificant Interaction
When the interaction is not significant,
interpretation of the main effects can
occur.
Main Effect and Interaction Means
Example
Use the following data to conduct a two-way
ANOVA
Group 1
Group 2
Group 3
4th Graders 14
14
10
10
13
12
11
15
14
5th Graders 14
14
15
11
11
10
13
14
15
Example
Summary Table
Sum of
Squares
Source
df
Mean
Square
F
Between
Factor A (Group)
23.444
2
11.722
5.146
Factor B (Grade)
0.889
1
0.889
0.390
Interaction
5.444
2
2.722
1.195
Within
27.333
12
2.278
Total
57.111
17
* Indicates significant at a = 0.05
*
Example
Since Factor A (Group) is significant…
k = 3 and n = 6, dfwn = 12, and a = .05, so q = 3.77
 MS wn
HSD  (qk )
n





 2.278 

 (3.77)
6 

 2.323
Example
X Group1  13.50
X Group 2  11.17
X Group3  13.67
• Since 13.5 – 11.17 = 2.33 is greater than 2.323, the
mean for Group 1 is significantly different from the
mean for Group 2
• Likewise, since 13.67 – 11.17 = 2.50 is greater than
2.323, the mean for Group 3 is significantly different
from the mean for Group 2
• But 13.67 – 13.50 = 0.17 is not greater than 2.323, so
the mean for Group 1 is not significantly different
from the mean for Group 3
Example
Effect size for Factor A (Group)
SS bn 23.444
h 

 0.4105
SStot 57.111
2
Example
Graphing the means
Group
16
14
12
10
8
6
4
2
0
1
2
3