Lecture 15: ANOVA Interactions Laura McAvinue School of Psychology Trinity College Dublin.
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Transcript of 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?
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
0
5
10
15
20
25
30
CBT Psycho-analytic
Drug
male
female
•Can’t say that one type of therapy is better for all clients
•Depends on gender
•Can’t say that one gender does better than the other
•Depends on therapy
•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?
CBT Psychoanalytic
Drug
Males 10 16 24
8 18 26
6 20 28
8 18 26
Females 22 6 20
20 4 22
18 8 24
20 6 22
Simple Effect 1: The effect of Gender under CBT condition
Which means do we compare?
CBT Psychoanalytic
Drug
Males 10 16 24
8 18 26
6 20 28
8 18 26
Females 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 Psychoanalytic
Drug
Males 10 16 24
8 18 26
6 20 28
8 18 26
Females 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 Psychoanalytic
Drug
Males 10 16 24
8 18 26
6 20 28
8 18 26
Females 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 Psychoanalytic
Drug
Males 10 16 24
8 18 26
6 20 28
8 18 26
Females 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?
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
216.000 1 216.000 54.000 .002
16.000 4 4.000
232.000 5
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
therapy = Psychoanalytica.
ANOVAa
depress
216.000 1 216.000 54.000 .002
16.000 4 4.000
232.000 5
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
therapy = CBTa.
ANOVAa
depress
24.000 1 24.000 6.000 .070
16.000 4 4.000
40.000 5
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
therapy = druga.
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
488.000 2 244.000 61.000 .000
24.000 6 4.000
512.000 8
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
gender = malea.
ANOVAa
depress
456.000 2 228.000 57.000 .000
24.000 6 4.000
480.000 8
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
gender = femalea.
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: depress
952.000a 5 190.400 47.600 .000
5000.000 1 5000.000 1250.000 .000
8.000 1 8.000 2.000 .183
496.000 2 248.000 62.000 .000
448.000 2 224.000 56.000 .000
48.000 12 4.000
6000.000 18
1000.000 17
SourceCorrected Model
Intercept
gender
therapy
gender * therapy
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .952 (Adjusted R Squared = .932)a.
Original MSerror = 4
ANOVAa
depress
216.000 1 216.000 54.000 .002
16.000 4 4.000
232.000 5
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
therapy = Psychoanalytica.
ANOVAa
depress
216.000 1 216.000 54.000 .002
16.000 4 4.000
232.000 5
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
therapy = CBTa.
ANOVAa
depress
24.000 1 24.000 6.000 .070
16.000 4 4.000
40.000 5
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
therapy = druga.
Simple Effects of Gender at all levels of Therapy
Simple Effects of Therapy at each level of Gender
ANOVAa
depress
488.000 2 244.000 61.000 .000
24.000 6 4.000
512.000 8
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
gender = malea.
ANOVAa
depress
456.000 2 228.000 57.000 .000
24.000 6 4.000
480.000 8
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
gender = femalea.
Original ANOVA table
Simple Effects 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
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