Transcript Factorial Design

Factorial Design
Main Effects and Interaction Effects
Pre-test
Determine the appropriateness of
manipulations (operationalizations of
independent variables)
 Test the validity and reliability of the
measurement scales

Pilot Study
Use a small group of participants to test
whether the questionnaire is clear and
understandable
 Use the pilot study data to provide some
intuitive observations about whether the
experiment could work

Factorial Design

Much experimental psychology asks the
question:
 What
effect does a single independent variable
have on a single dependent variable?

It is quite reasonable to ask the following
question as well.
 What
effects do multiple independent variables
have on a single dependent variable?

Designs which include multiple independent
variables are known as factorial designs.
Factorial Design



All combinations of two or more values of two or
more IVs are created
Can be tested within, between or mixed
Reasons for factorial designs
– additional Ivs are included
 Helps rule out several hypotheses at the same time
 Allows the researcher to discover interactions
amongst the variables
 Allows the researcher to include one or more
nuisance variables in the design
 Efficiency
Picture of Toy Factorial Design
Bright colour
Dull Colour
Not Noisy
Noisy
Soft
Not Soft
Describing Toy Factorial Design
All possible combinations of soft, noisy and colourful
toys in a 2 x 2 x 2 design – i.e., three IVs
2 x 2 x 2 = 8 cells
Another factor (IV) with two levels
One factor (IV) with 2 levels
An Example Factorial Design

If we were looking at GENDER and TIME
OF EXAM, these would be two
independent factors
 GENDER
would only have two levels: male or
female
 TIME OF EXAM might have multiple levels,
e.g. morning, noon or night

This is a factorial design
Examples

If there are 2 levels of the first IV and 3 levels of
the second IV
 It
is a 2x3 design

E.G.: coffee drinking x time of day



Factor coffee has two levels: cup of coffee or cup of water
Factor time of day has three levels: morning, noon and night
If there are 3 levels of the first IV, 2 levels of the
second IV and 4 levels of the third IV
 It
is a 3x2x4 design

E.G.: coffee drinking x time of day x exam duration



Factor coffee has three levels: 1 cup, 2 cup 3 cups
Factor time of day has two levels: morning or night
Factor exam duration has 4 levels: 30min, 60min, 90min,
120min
Difficulties with Factorial Designs
More complex, more time
 Difficult to analyze if some data is missing
 Need more Ss for each factor, for each
extra level
 Interactions, especially higher order
interactions (3, 4 or higher), can be difficult
to understand

Main Effects and Interaction Effects
The effect of a single variable is known as a
main effect
 The effect of two variables considered
together is known as an interaction
 For the two-way between groups design, an
F-ratio is calculated for each of the
following:

 The
main effect of the first variable
 The main effect of the second variable
 The interaction between the first and second
variables
Analysis of a 2-way BetweenSubjects Design Using ANOVA

To analyse the two-way between groups
design we have to follow the same steps
as the one-way between groups design
 State
the Null Hypotheses
 Partition the Variability
 Calculate the Mean Squares
 Calculate the F-Ratios
Null Hypotheses

There are 3 null hypotheses for the twoway (between groups design.
 The
means of the different levels of the first IV
will be the same, e.g.  A   A   A
1
2
 The
means of thedifferent levels of the
second IV will be the same, e.g.  B   B   B
1
2
 means of the
differences between the
different levels of the interaction are not the
same, e.g.  AB   AB   AB   AB
 The
11
21
21
22
An Example Null Hypothesis for an
Interaction
140
120
100
Lectures
80
No Lectures
60
40
20
0
Worksheets
No Practice
 AB11   AB21   AB21   AB22

The differences betweens the levels of
factor A are not the same.
Partitioning the Variability

If we consider the different levels of a oneway ANOVA then we can look at the
deviations due to the between groups
variability and the within groups variability.
ASij  T  (Ai  T )(ASij  Ai )

If wesubstitute AB into the above equation
we get ABS  T  (AB  T )  (ABS  AB )
ijk

ij
ij
ij
This provides the deviations associated
with between and within groups variability
for the two-way between groups design.
Partitioning the Variability (Cont.)

The between groups deviation can be
thought of as a deviation that is comprised
of three effects.
(AB ij  T )  Ai effect B j effect Ai xB j effect

In other words the between groups

variability
is due to the effect of the first
independent variable A, the effect of the
second variable B, and the interaction
between the two variables AxB.
Partitioning the Variability

The effect of A is given by

Similarly the effect of Bis given by

The effect of the interaction AxBequals

which is known as a residual
Ai  T
Bj  T
( ABij  T )  ( Ai  T )  ( Bk  T )
The Sum of Squares

The sums of squares associated with the twoway between groups design follows the same
form as the one-way
SST  SSA  SSB  SSAB  SSS / AB


We need to calculate a sum of squares
associatedwith the main effect of A, a sum of
squares associated with the main effect of B, a
sum of squares associated with the effect of the
interaction.
From these we can estimate the variability due
to the two variables and the interaction and an
independent estimate of the variability due to the
error.
The Mean Squares

In order to calculate F-Ratios we must
calculate an Mean Square associated with
 The
Main Effect of the first IV
 The Main Effect of the second IV
 The Interaction.
 The Error Term
The mean squares

The main effect mean squares are given
SS
by: MS  SS where df  a 1
MS 
where df  b 1
A
A
df A
MSAB 


A
B
dfB
B
SSAB
where df AB  (a 1)(b 1)
df AB

The interaction mean squares is given by:

MSS / AB 

B
SSS / AB
where dfS / AB  ab(s 1)
dfS / AB
The error mean square is given by:
The F-ratios

The F-ratio for the first main effect is:
MSA
MSS / AB

The F-ratio forthe second main effect is:
MSB
MSS / AB

The F-ratio forthe interaction is:
MSAB
MSS / AB
An Example 2x2 Between-Groups
ANOVA
Factor A - Lectures (2 levels: yes, no)
 Factor B - Worksheets (2 levels: yes, no)
 Dependent Variable - Exam performance
(0…30)

Mean
Std Error
LECTURES
WORKSHEETS
yes
yes
19.200
2.04
no
25.000
1.23
yes
16.000
1.70
no
9.600
0.81
no
Results of ANOVA
• When an analysis of variance is conducted on
the data, the following results are obtained
Source
Sum of
Squares
df
Mean
F
Squares
p
A (Lectures)
432.450
1
432.450
37.604
0.000
B (Worksheets)
0.450
1
0.450
0.039
0.846
AB
186.050
1
186.050
16.178
0.001
Error
184.000
16
11.500
What Does It Mean? - Main effects

A significant main effect of Factor A
(lectures)
 “There
was a significant main effect of lectures
(F1,16=37.604, MSe=11.500, p<0.001). The
students who attended lectures on average
scored higher (mean=22.100) than those who
did not (mean=12.800).

No significant main effect of Factor B
(worksheets)
 “The
main effect of worksheets was not
significant (F1,16=0.039, MSe=11.500, p=0.846)”
What Does It Mean? - Interaction

A significant interaction effect
 “There
was a significant interaction between the
lecture and worksheet factors (F1,16=16.178,
MSe=11.500, p=0.001)”

However, we cannot at this point say anything
specific about the differences between the
means unless we
look
at the null
hypothesis



AB11
LECTURES

yes
no

AB21
AB21
AB22
Mean
Std Error
yes
19.200
2.04
no
25.000
1.23
yes
16.000
1.70
no
9.600
0.81
WORKSHEETS
Many researches prefer to continue to make
more specific observations.
Analytic Comparisons in General
If there are more than two levels of a
Factor
 And, if there is a significant effect (either
main effect or simple main effect)

 Analytical
comparisons are required.
 Post hoc comparisons include Tukey tests,
Scheffé test or t-tests (Bonferroni corrected).

2X2
Independent Variable A

3X3X2
B1
Independent Variable
C
A1
Independent
Variable A
A2
A3

A1
A2
C1
Independent Variable B
B1
B2
A1B1
A1 B2
A2B1
A2 B2
Independent Variable B
B2
C2
C1
C2
C1

C2
A1B1C
A1B1C
A1B2C
A1B2C
A1B3C
A1B3C
1
2
1
2
1
2
A2B1C
A2B1C
A2B2C
A2B2C
A2B3C
A2B3C
1
2
1
2
1
2
A3B1C
A3B1C
A3B2C
A3B2C
A3B3C
A3B3C
1
2
1
2
1
2
Assignment to conditions

B3
Randomized group—completely randomized factorial
Matched factorial design—form groups based on
matching and randomly assign to conditions
The Purpose of Factorial Models
Two or more variables at once
 Combinations of variables
 Main effects

 The
effect of one independent variable,
averaged over all levels of another
independent variable
 The effect of one independent variable,
ignoring the other independent variables

Interaction
 effect
of one independent variable
depends upon the levels of another
independent variable
 the effects of one variable varies over
the levels of another independent
variable

Main effects
 Model
 Stimulus

R
Group A
T frightened model
R
Group B
T not frightened
model
R
Group A
R
Group B
T plastic
snake
T plastic
flower
M emotional
response
M emotional
response
M emotional
response
M emotional
response
Interaction
Stimulus
Plastic flower
Plastic snake
Model Emotion
Frightened
Not frightened
Group A
Group C
Group B
Group D

Three hypotheses
 Main
effects
Stimulus
 Model emotion

 Interaction—Stimulus
X model
emotion

Actual interaction found
Stimulus

Plastic flower
Plastic snake
Model Emotion
Frightened
Not frightened
No Fright
No Fright
Fright
No Fright
Results from similar study
Stimulus
Plastic flower
Plastic snake
Model Emotion
Frightened
Not frightened
.1
0
1.2
.1
Interactions

Graphic representation
Model
Frightened
Not Frightened
Response
1.00
0.75
0.50
0.25
0.00
Flower
Snake
Stimulus

Main effects conditioned by interaction
Mood and Memory
Recall Mood
Sad
Happy
Recall Mood Sad
Learning Mood
Sad
Happy
.8
.46
.45
.78
Recall Mood Happy
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
Sad
Happy
Learning Mood
Factorial Design Analysis

2X2


Three questions
 Two main effects
 Interaction
Source table






Source
SS
df
MS
F
Significance
f
B
3
1
t
w
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e
D
T
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1
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
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p
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. 89 S
N
. 60
. 77
. 07
. 68
0M
S
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S
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0
4 33
0 85
1
3 60
8 73
1
8 06
9 53
3
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0
t
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1
9
3S
n
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3 33
9 79
51
3 70
0 22
51
8 06
9 62
03
re
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d
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d
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9 6
2
No effects
1
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5
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Estimated Marginal Means
.
3
Possible Patterns
14.0
12.0
10.0
Strength of Argument
8.0
Weak
6.0
Strong
High
Low
Involvement in issue (own school versus other school)
Main Effect No Interaction
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Estimated Marginal Means of PERSUAS2
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1
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0
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Estimated Marginal Means
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2
1
5
18.0
N
5
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4
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5
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3
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5
9
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12.0
N
0
0
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Strength of Argument
10.0
Weak
8.0
Strong
High
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Involvement in issue (own school versus other school)
Two Main Effects No Interaction
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Estimated Marginal Means
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5
5
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3
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5
3
5
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0
0
0
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Strength of Argument
8.0
Weak
6.0
Strong
High
Low
Involvement in issue (own school versus other school)
Interaction Only
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7
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Estimated Marginal Means
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6
3
5
N
5
5
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18.0
2
M
.
0
3
7
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0
9
8
16.0
N
5
5
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3
0
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14.0
S
3
8
7
N
0
0
0
12.0
Strength of Argument
10.0
Weak
8.0
Strong
High
Low
Involvement in issue (own school versus other school)
Main Effect and Interaction
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Estimated Marginal Means
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7
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18.0
Estimated Marginal Means
S
6
5
6
N
5
5
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16.0
2
M
.
7
0
3
S
1
8
4
14.0
N
5
5
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M
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7
0
3
12.0
S
6
0
2
N
0
0
0
10.0
Strength of Argument
8.0
Weak
6.0
Strong
High
Low
Involvement in issue (own school versus other school)
Two Main effects and Interaction
e
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D
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1
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6
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7
1
7
1
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1
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2
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3
5
1
3
6
0
T
0
6 0
0
e
C
8
5 3
9
D
e
a
R
N
eI
n
g
i
s
s
o
S
W
t
a
t
Estimated Marginal t
Means of PERSUAS2
o
1
M
.
0
3
7
Estimated Marginal Means
20.0
S
2
2
1
N
5
5
0
18.0
2
M
.
0
0
0
S
8
7
1
16.0
N
5
5
0
T
M
o
0
7
3
14.0
S
0
3
7
N
0
0
0
12.0
10.0
Strength of Argument
8.0
Weak
6.0
Strong
High
Low
Involvement in issue (own school versus other school)
Interactions

Interactions
 Antagonistic
 Synergistic
 Ceiling
effect
Higher Order Designs

Extensions within a two way design
 More
3

levels 2 X 3
X3
Higher order designs
 More

factors
2X2X2
 Limitations
Interpretation of higher order interactions
 Number of conditions and participants


3 X 3 X 3 X 3 = 81 cells
Higher order interactions
Involvement in
issue
High
Low
Strength of arguments
Weak
Strong
Number of
Number of
arguments
arguments
3
9
3
9
4.10
1.05
8.32
11.3
4.52
7.71
4.95
8.66
Low Involvement
10
9
8
7
6
Weak Arguments
5
Strong Arguments
4
3
2
1
0
3
9
Number of Arguments
High Involvement
12
10
8
Weak
Strong
6
4
2
0
3
9
Number of Arguments
What Is Multivariate Analysis of
Variance (MANOVA)?

Univariate Procedures For Assessing
Group Differences
 The
t Test
 Analysis of Variance (ANOVA)

Multivariate Analysis of Variance
(MANOVA)
 The
Two-Group Case: Hotelling's T2
 The k-Group Case: MANOVA
Group Comparison Analyses
ANOVA
 ANCOVA
 MANOVA
 MANCOVA
