Lecture 15:
ANOVA Interactions
Laura McAvinue
School of Psychology
Trinity College Dublin
Factorial ANOVA
• Two or more independent variables
• Allows us to examine two kinds of ‘effect’. What are
these?
• Main effects
• The effect of each independent variable, controlling for the
other variable
• Simple effects
• Interaction between the independent variables
• The effect of one independent variable at one level of
another variable
Recall our research example
• We would like to examine the effectiveness of three kinds
of therapy (CBT, psychoanalytic, drug) on depressive
symptoms displayed by men & women
• This design will enable us to investigate three things
What are these?
– Main effect of Gender
– Main effect of Therapy
– Interaction between Gender & Therapy
Recall our research example
• Examine the following graphs of possible results
for our study and for each one tell me…
– Is there a main effect of Gender?
– Is there a main effect of Therapy?
– Is there an interaction between Gender & Therapy?
Main effect of Gender
Interaction
Main effect of Therapy
Interaction
Main effect of Therapy
No effect of Gender or
Therapy
Main effect of Gender &
Therapy
Interaction
Graphs of Interactions
• No interaction
– Lines are parallel
• Interaction
– Lines are not parallel
– Lines cross or look like they might cross if the graph
was extrapolated
• Is the interaction significant?
– ANOVA, significance of F value
Interactions
• The independent variables have a combined effect on the
dependent variable
• The effects of one variable differ at different levels of the
other variable
• Renders a main effect less important
• Often, if there is an interaction, you should focus on this
rather than on the main effects
Interactions
• So, you have found a significant interaction between the
independent variables…
• But what kind of interaction is it?
• Examine graph
• Analysis of Simple Effects
– Factorial ANOVA enables you to pair each level of one variable
with every level of the other variable
– Analysis of simple effects allows you to tease apart the interaction
– …allows you to compare the pairings to see where the interaction
lies
Simple Effects
• The effect of one variable at just one level of a second
variable
• Involves running several One Way ANOVAs
• You exclude certain parts of the data and just examine
the parts you are interested in
• There are often many simple effects that you can analyse
– But you increase the risk of making a Type I error
– Usually, go by the graph and only analyse the simple effects that
you think are important
Our Research Example
30
•Can’t say that one
type of therapy is
better for all clients
25
20
•Depends on
gender
15
10
male
female
5
•Can’t say that one
gender does better
than the other
•Depends on
therapy
0
CBT
Psychoanalytic
Drug
•Need to consider
both gender &
therapy when
interpreting data
Simple Effects
• We need to examine the effects of Gender at all levels of
Therapy
• &
• The effects of Therapy at all levels of Gender
• This gives us 5 simple effects to analyse
• What are these?
Simple Effects
• The effects of Gender at all levels of Therapy
– The effect of gender in CBT condition
• Do men & women receiving CBT differ?
– The effect of gender in psychoanalysis condition
• Do men & women receiving psychoanalysis differ?
– The effect of gender in drug condition
• Do men & women receiving drugs differ?
• The effects of Therapy at each level of Gender
– The effect of therapy for males
• Is at least one therapy mean significantly different from the others for
males?
– The effect of therapy for females
• Is at least one therapy mean significantly different from the others for
females?
Simple Effect 1: The effect of Gender under CBT condition
Which means do we compare?
CBT
Males
Females
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
8
18
26
22
6
20
20
4
22
18
8
24
20
6
22
Simple Effect 2: The effect of gender under psychoanalysis
condition
Which means do we compare?
CBT
Males
Females
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
8
18
26
22
6
20
20
4
22
18
8
24
20
6
22
Simple Effect 3: The effect of gender under drug condition
Which means do we compare?
CBT
Males
Females
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
8
18
26
22
6
20
20
4
22
18
8
24
20
6
22
Simple Effect 4: The effect of therapy for males
Which means do we compare?
CBT
Males
Females
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
8
18
26
22
6
20
20
4
22
18
8
24
20
6
22
Simple Effect 5: The effect of therapy for females
Which means do we compare?
CBT
Males
Females
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
8
18
26
22
6
20
20
4
22
18
8
24
20
6
22
Simple Effects on SPSS
• Not easy to do on SPSS
• To examine the effects of gender at all levels of therapy…
– Split file
– Organise output according to therapy
– One Way ANOVA with gender as independent variable &
depression as dependent variable
• Output will produce three One Way ANOVAs
– The effects of gender on depression under CBT condition
– The effects of gender on depression under psychoanalysis
– The effects of gender on depression under drug condition
Simple effects of gender at each level of
therapy
ANOVAa
depress
Between Groups
Within Groups
Total
Sum of
Squares
216.000
16.000
232.000
df
1
4
5
Mean Square
216.000
4.000
F
54.000
Sig.
.002
F
54.000
Sig.
.002
F
6.000
Sig.
.070
a. therapy = Ps ychoanalytic
ANOVAa
depress
Between Groups
Within Groups
Total
Sum of
Squares
216.000
16.000
232.000
df
1
4
5
Mean Square
216.000
4.000
a. therapy = CBT
ANOVAa
depress
Between Groups
Within Groups
Total
a. therapy = drug
Sum of
Squares
24.000
16.000
40.000
df
1
4
5
Mean Square
24.000
4.000
Simple Effects on SPSS
• To examine the effects of therapy at each level of
gender…
– Split file
– Organise output according to gender
– One Way ANOVA with therapy as independent variable &
depression as dependent variable
• Output will produce two One Way ANOVAs
– The effects of therapy on depression for males
– The effects of therapy on depression for females
Simple effects of therapy at each level of
gender
ANOVAa
depress
Between Groups
Within Groups
Total
Sum of
Squares
488.000
24.000
512.000
df
2
6
8
Mean Square
244.000
4.000
F
61.000
Sig.
.000
F
57.000
Sig.
.000
a. gender = male
ANOVAa
depress
Between Groups
Within Groups
Total
Sum of
Squares
456.000
24.000
480.000
a. gender = female
df
2
6
8
Mean Square
228.000
4.000
Create a new ANOVA table
• By hand!
• Take the average variation (MS) due to each of your
simple effects
• Create a new ANOVA table
– using these MS & the old MSerror term
• Compute F ratio for each simple effect by comparing the
MS for each simple effect to the original MSerror term
– Look up the probability of obtaining this F ratio when Ho is true,
using the F distribution table
The original ANOVA
Tests of Between-Subjects Effects
Dependent Variable: depres s
Source
Corrected Model
Intercept
gender
therapy
gender * therapy
Error
Total
Corrected Total
Type III Sum
of Squares
952.000 a
5000.000
8.000
496.000
448.000
48.000
6000.000
1000.000
df
5
1
1
2
2
12
18
17
Mean Square
190.400
5000.000
8.000
248.000
224.000
4.000
a. R Squared = .952 (Adjusted R Squared = .932)
Original MSerror = 4
F
47.600
1250.000
2.000
62.000
56.000
Sig.
.000
.000
.183
.000
.000
Simple Effects of Gender
at all levels of Therapy
Simple Effects of Therapy
at each level of Gender
ANOVAa
depress
Between Groups
Within Groups
Total
Sum of
Squares
216.000
16.000
232.000
ANOVAa
df
1
4
5
Mean Square
216.000
4.000
F
54.000
Sig.
.002
depress
Between Groups
Within Groups
Total
a. therapy = Ps ychoanalytic
Sum of
Squares
488.000
24.000
512.000
df
2
6
8
Mean Square
244.000
4.000
F
61.000
Sig.
.000
F
57.000
Sig.
.000
a. gender = male
ANOVAa
depress
Between Groups
Within Groups
Total
Sum of
Squares
216.000
16.000
232.000
df
1
4
5
Mean Square
216.000
4.000
F
54.000
Sig.
.002
ANOVAa
depress
a. therapy = CBT
Between Groups
Within Groups
Total
a
ANOVA
depress
Between Groups
Within Groups
Total
a. therapy = drug
Sum of
Squares
24.000
16.000
40.000
df
1
4
5
Mean Square
24.000
4.000
F
6.000
Sig.
.070
Sum of
Squares
456.000
24.000
480.000
a. gender = female
df
2
6
8
Mean Square
228.000
4.000
Original ANOVA table
Source
SS
Df
MS
F
Pvalue
Gender
8
1
8
2
.183
Therapy
496
2
248
62
.000
Gender*Therapy
448
2
224
56
.000
Error
48
12
4
Simple Effects ANOVA table
Source
SS
Df
MS
F
Critical F Signif?
Gender at CBT
216
1
216
54
4.75
Yes
Gender at Psycho
216
1
216
54
4.75
Yes
Gender at Drug
24
1
24
6
4.75
Yes
Therapy at males
448
2
224
56
3.89
Yes
Therapy at females
456
2
228
57
3.89
Yes
Error
48
12
4
What can we conclude?
• No main effect of gender
• Main effect of therapy
• Interaction between
gender & therapy
• All simple effects are
statistically significant