1. Main Effects and Interaction Effects
Factorial Design
Main Effects and Interaction Effects

2. Pre-test
Determine the appropriateness of manipulations (operationalizations of independent variables)
Test the validity and reliability of the measurement scales

3. 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

4. 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.

5. 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
Efficiency – 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

6. Picture of Toy Factorial Design
Bright colour
Dull Colour
Not Noisy
Noisy
Soft
Not Soft

7. 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

8. 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

9. 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

10. 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

11. 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

12. Analysis of a 2-way Between-Subjects 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

13. Null Hypotheses
There are 3 null hypotheses for the two-way (between groups design.
The means of the different levels of the first IV will be the same, e.g.
The means of the different levels of the second IV will be the same, e.g.
The differences between the means of the different levels of the interaction are not the same, e.g.

14. An Example Null Hypothesis for an Interaction
The differences betweens the levels of factor A are not the same.

15. Partitioning the Variability
If we consider the different levels of a one-way ANOVA then we can look at the deviations due to the between groups variability and the within groups variability.
If we substitute AB into the above equation we get
This provides the deviations associated with between and within groups variability for the two-way between groups design.

16. Partitioning the Variability (Cont.)
The between groups deviation can be thought of as a deviation that is comprised of three effects.
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.

17. 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

18. The Sum of Squares
The sums of squares associated with the two-way between groups design follows the same form as the one-way
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.

19. 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

20. The mean squares The main effect mean squares are given by:
The interaction mean squares is given by:
The error mean square is given by:

21. The F-ratios The F-ratio for the first main effect is:
The F-ratio for the second main effect is:
The F-ratio for the interaction is:

22. 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
19.200
2.04 no
25.000
1.23
16.000
1.70
9.600
0.81

23. Results of ANOVA
When an analysis of variance is conducted on the data, the following results are obtained
Source
Sum of Squares df
Mean Squares F p
A (Lectures) 1
37.604
0.000
B (Worksheets)
0.450
0.039
0.846 AB
16.178
0.001
Error 16
11.500

24. 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)”

25. 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
Many researches prefer to continue to make more specific observations.
Mean
Std Error
LECTURES
WORKSHEETS
yes
19.200
2.04 no
25.000
1.23
16.000
1.70
9.600
0.81

26. 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).

27. Assignment to conditions
2 X 2
3 X 3 X 2
Assignment to conditions
Randomized group—completely randomized factorial
Matched factorial design—form groups based on matching and randomly assign to conditions

28. 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

29. Interaction Main effects
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

30. Actual interaction found
Three hypotheses
Main effects
Stimulus
Model emotion
Interaction—Stimulus X model emotion
Actual interaction found
Results from similar study

31. Interactions Graphic representation
Main effects conditioned by interaction

32. Mood and Memory

33. Factorial Design Analysis
2 X 2
Three questions
Two main effects
Interaction
Source table
Source SS df MS F
Significance

34. Possible Patterns
No effects

35. Main Effect No Interaction

36. Two Main Effects No Interaction

37. Interaction Only

38. Main Effect and Interaction

39. Two Main effects and Interaction

40. Interactions
Interactions
Antagonistic
Synergistic
Ceiling effect

41. Higher Order Designs Extensions within a two way design
More levels 2 X 3
3 X 3
Higher order designs
More factors
2 X 2 X 2
Limitations
Interpretation of higher order interactions
Number of conditions and participants
3 X 3 X 3 X 3 = 81 cells

42. Higher order interactions

44. 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

45. Group Comparison Analyses
ANOVA
ANCOVA
MANOVA
MANCOVA