Transcript Lecture 11: One Way ANOVA Repeated Measures

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
Between
Repeated
subjects
measures /
Within
subjects
Different
participants
Same
participants
Two
way
Three
way
Four
way
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
Type of Therapy
CBT
Gender
Males
Females
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
22
6
20
20
4
22
18
8
24
Is there a main effect of Gender?
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
17.33
16
16.67
Is there a main effect of Therapy?
CBT
Males
Females
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
22
6
20
20
4
22
18
8
24
14
12
24
16.67
Is there an Interaction between Gender & Therapy?
Examine the pattern of means…
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
16.67
Line graph of the six cell means
30
25
male
20
female
15
10
5
0
CBT
Psychoanalytic
Drug
Calculations
Total Variance
Variance
Variance
Variance
Variance
due to IV1
due to IV2
Due to
Gender
Therapy
due to the
interaction
between
IV1 & IV2
Gender x
Therapy
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
Total Variance
CBT
Males
Females
SStotal
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
22
6
20
20
4
22
18
8
24
∑ (xij - Grand Mean )2
16.67
1000
Variance due to Gender
CBT
Males
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
17.33
Females
22
6
20
20
4
22
18
8
24
16
16.67
SSgender
ngender ∑ (Mean for each level of gender - Grand Mean )2
Variance due to Gender
CBT
Males
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
17.33
Females
22
6
20
20
4
22
18
8
24
16
16.67
SSgender
9 ∑ (17.33 – 16.67 )2 + (16 – 16.67)2
8
Variance due to Therapy
CBT
Males
Females
SStherapy
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
22
6
20
20
4
22
18
8
24
14
12
24
16.67
ntherapy ∑ (Mean for each level of therapy - Grand Mean )2
Variance due to Therapy
CBT
Males
Females
SStherapy
Psychoanaly Drug
tic
10
16
24
8
18
26
6
20
28
22
6
20
20
4
22
18
8
24
14
12
24
16.67
6 ∑ (14 – 16.67 )2 + (12 – 16.67)2 + (24 – 16.67)2
496
Variance due to the interaction
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
16.67
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
Variance due to the interaction
CBT
Males
Females
SScells
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
16.67
No. of participants in each cell ∑ (Each cell mean - Grand Mean )2
Variance due to the interaction
CBT
Males
Females
SScells
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
16.67
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
952
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 MSgender*therapy P value
/ dfgender*therapy
MSerror
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