Lecture 13:Factorial ANOVA 1

Laura McAvinue

School of Psychology

Trinity College Dublin

Analysis of Variance

One way ANOVA Factorial ANOVA

One Independent Variable

More than One Independent Variable

Two way

Three way

Four way

Between

subjects

Repeated

measures /

Within

subjects

Different participants

Same participants

Factorial ANOVA

• Factor– Another word for an independent variable in ANOVA

• Factorial Design– Design in which there are two or more independent

variables or factors

Labelling

• Number of independent variables / factors

– One independent variable – One way ANOVA– Two independent variables – Two way ANOVA– Three independent variables – Three way ANOVA

• Number of levels of each variable / factor– Comparing men and women’s performance on an attention task

under three conditions of noise– Two independent variables

• Gender (2 levels: male & female)• Noise (3 levels: none, white noise, random tones)

– 2 x 3 factorial ANOVA

Factorial ANOVA

• Allows you to examine two things…

– The main effect of each independent variable, when controlling for the other variable

– The interaction between the two variables

Research Example

• We would like to examine the effectiveness of three kinds of therapy (CBT, psychoanalytic, drug) on depressive symptoms displayed by men & women

• What is our dependent variable?– Number of depressive symptoms

• How many independent variables do we have?– 2

Research Example

• We would like to examine the effectiveness of three kinds of therapy (CBT, psychoanalytic, drug) on depressive symptoms displayed by men & women

• What are the independent variables?– Gender & Therapy

• How many levels do they have?– Gender: 2 levels (Male/Female)– Therapy: 3 levels (CBT, Psychoanalytic, Drug)

Research Example

• We would like to examine the effectiveness of three kinds of therapy (CBT, psychoanalytic, drug) on depressive symptoms displayed by men & women

• Label this experiment in two ways– Two Way Factorial ANOVA– 2 x 3 Factorial ANOVA

Factorial ANOVA

• This design will enable us to investigate three things

– Main Effect of Gender

– Main Effect of Therapy

– Interaction between Gender and Therapy

Main Effect

• The effect of one independent variable averaged across the levels of the other independent variable

• The effect of one independent variable ignoring the other variable

Main Effect of Gender

• There is a significant difference between men and women’s no. of depressive symptoms across all therapy groups– Men and women’s depressive symptoms differ, irrespective of

the type of therapy they got– The type of therapy does not influence the effect of gender

• E.g. Men have a significantly lower number of depressive symptoms than women overall, across all three therapy conditions– Ho: There is no effect of gender

• Mean of males = Mean of females

– Halt: There is a main effect of gender

• Mean of males ≠ Mean of females

Main Effect of Therapy

• The kind of therapy administered significantly affected the number of depressive symptoms, irrespective of the gender of the client

• Ho: There is no significant effect of therapy– Mean CBT = Mean Psychoanalytic = Mean Drug

• Halt: At least one mean for therapy is different from the other two

• E.g. CBT significantly reduced the number of depressive symptoms for both men and women

Interaction

• Factorial Design– Enables you to pair each level of each variable with each level of

the other variable / variables

• Interaction– Combined effect IV1 & IV2 on the DV

– Means that the effects of one independent variable depend on the level of the other independent variable

• Simple Effect– The effect of one independent variable at one level of another

variable

Interaction between Gender & Therapy

• One therapy is more effective for one type of client

• Men & women benefit equally from CBT and drugs but women respond better to psychoanalysis

• Ho: There is no interaction between gender & therapy– All mean differences are due only to main effects

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

Females 22 6 20

20 4 22

18 8 24

Type of TherapyG

ende

r

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

8 18 26 17.33

Females 22 6 20

20 4 22

18 8 24

20 6 22 16

16.67

Is there a main effect of Gender?

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

Females 22 6 20

20 4 22

18 8 24

14 12 24 16.67

Is there a main effect of Therapy?

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

16.67

Is there an Interaction between Gender & Therapy?

Examine the pattern of means…

Line graph of the six cell means

0

5

10

15

20

25

30

CBT Psycho-analytic

Drug

male

female

Calculations

Total Variance

Variance

due to IV1

Gender

Variance

due to IV2

Therapy

Variance

due to the interaction between

IV1 & IV2

Gender x Therapy

Variance

Due to

random

error

Three F Ratios

Compare the variance due to the main effects and the interaction to the variance due to random error

Variance due to Gender

Variance due to Random Error

Variance due to Therapy

Variance due to Random Error

Variance due to Gender x Therapy

Variance due to Random Error

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

Females 22 6 20

20 4 22

18 8 24 16.67

Total Variance

SStotal∑ (xij - Grand Mean )2

1000

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

17.33Females 22 6 20

20 4 22

18 8 24

16

16.67

Variance due to Gender

SSgenderngender ∑ (Mean for each level of gender - Grand Mean )2

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

17.33Females 22 6 20

20 4 22

18 8 24

16

16.67

Variance due to Gender

SSgender9 ∑ (17.33 – 16.67 )2 + (16 – 16.67)2

8

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

Females 22 6 20

20 4 22

18 8 24

14 12 24 16.67

Variance due to Therapy

SStherapyntherapy ∑ (Mean for each level of therapy - Grand Mean )2

CBT Psychoanalytic

Drug

Males 10 16 24

8 18 26

6 20 28

Females 22 6 20

20 4 22

18 8 24

14 12 24 16.67

Variance due to Therapy

SStherapy6 ∑ (14 – 16.67 )2 + (12 – 16.67)2 + (24 – 16.67)2

496

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 16.67

Variance due to the interaction

Each cell mean is a combination of a level of each independent variable

Variance due to the Interaction

• SScells

– The sum of squared deviations of each cell mean from the grand mean

– The variance of the cell means– A measure of how much the cell means differ

• Cell means can differ due to…– Level of Gender– Level of Therapy– Interaction between Gender & Therapy

Variance due to the Interaction

• SScells = SSgender + SStherapy + SSgender x therapy

• SSgender x therapy = SScells – SSgender – SStherapy

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 16.67

Variance due to the interaction

No. of participants in each cell ∑ (Each cell mean - Grand Mean )2

SScells

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 16.67

Variance due to the interaction

3 ∑ (8 – 16.67 )2 + (18 – 16.67)2 + (26 – 16.67)2 + (20 – 16.67)2 + (6 – 16.67)2 + (22 – 16.67)2

952SScells

Variance due to the Interaction

• SSgender x therapy = SScells – SSgender – SStherapy

• SSgender x therapy = 952 – 8 – 496

• SSgender x therapy = 448

Variance due to Random Error

• Two Methods…

• Directly– SSerror = ∑(each score in each cell – mean of that cell)2

– 48

• Indirectly– SStotal = [SSgender + SStherapy + SSgender x therapy ] + SSerror

– SStotal = [SScells ] + SSerror

– SSerror = SStotal – SScells

= 1000 – 952 = 48

ANOVA table

Source of variation

SS df MS F p

Gender SSgender kgender – 1 SSgender / dfgender

MSgender

MSerror

P value

Therapy SStherapy ktherapy – 1 SStherapy / dftherapy

MStherapy

MSerror

P value

Gender*

Therapy

SSgender*therapy Dfgender * Dftherapy

SSgender*therapy / dfgender*therapy

MSgender*therapy

MSerror

P value

Error SSerror kgender * ktherapy

(n-1)

SSerror / dferror

Total SStotal N – 1

ANOVA table

Source of variation

SS df MS F p

Gender 8 1 8 2 .183

Therapy 496 2 248 62 .000

Gender*

Therapy

448 2 224 56 .000

Error 48 12 4

Total 1000 17