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

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

What can we conclude?

• No main effect of gender

• Main effect of therapy

• Interaction between gender & therapy

• All simple effects are statistically significant