1. Statistics for the Social Sciences
Psychology 340
Spring 2010
Factorial ANOVA

2. Outline Basics of factorial ANOVA Interpretations Computations
Main effects
Interactions
Computations
Assumptions, effect sizes, and power
Other Factorial Designs
More than two factors
Within factorial ANOVAs
Mixed factorial ANOVAs

3. Statistical analysis follows design
The factorial (between groups) ANOVA:
More than two groups
Independent groups
More than one Independent variable

4. Factorial experimentsB1 B2 B3 A1 A2
Two or more factors
Factors - independent variables
Levels - the levels of your independent variables
2 x 3 design means two independent variables, one with 2 levels and one with 3 levels
“condition” or “groups” is calculated by multiplying the levels, so a 2x3 design has 6 different conditions

5. Factorial experiments
Two or more factors (cont.)
Main effects - the effects of your independent variables ignoring (collapsed across) the other independent variables
Interaction effects - how your independent variables affect each other
Example: 2x2 design, factors A and B
Interaction:
At A1, B1 is bigger than B2
At A2, B1 and B2 don’t differ

6. Results So there are lots of different potential outcomes:
A = main effect of factor A
B = main effect of factor B
AB = interaction of A and B
With 2 factors there are 8 basic possible patterns of results:
1) No effects at all
2) A only
3) B only
4) AB only
5) A & B
6) A & AB
7) B & AB
8) A & B & AB

7. 2 x 2 factorial design Interaction of AB A1 A2 B2 B1 Marginal means
What’s the effect of A at B1?
What’s the effect of A at B2?
Condition
mean
A1B1
Condition
mean
A2B1
Marginal means
B1 mean
B2 mean
A1 mean
A2 mean
Main effect of B
Condition
mean
A1B2
Condition
mean
A2B2
Main effect of A

8. Examples of outcomes Main effect of A √ Main effect of B
Dependent Variable B1 B2 30 60 45 60 45 30 30 60
Main Effect
of A
Main effect of A √
Main effect of B X
Interaction of A x B X

9. Examples of outcomes Main effect of A Main effect of B √
Dependent Variable B1 B2 60 60 60 30 30 30 45 45
Main Effect
of A
Main effect of A X
Main effect of B √
Interaction of A x B X

10. Examples of outcomes Main effect of A Main effect of B
Dependent Variable B1 B2 60 30 45 60 45 30 45 45
Main Effect
of A
Main effect of A X
Main effect of B X
Interaction of A x B √

11. Examples of outcomes Main effect of A √ Main effect of B √
Dependent Variable B1 B2 30 60 45 30 30 30 30 45
Main Effect
of A
Main effect of A √
Main effect of B √
Interaction of A x B √

12. Factorial Designs
Benefits of factorial ANOVA (over doing separate 1-way ANOVA experiments)
Interaction effects
One should always consider the interaction effects before trying to interpret the main effects
Adding factors decreases the variability
Because you’re controlling more of the variables that influence the dependent variable
This increases the statistical Power of the statistical tests

13. Basic Logic of the Two-Way ANOVA
Same basic math as we used before, but now there are additional ways to partition the variance
The three F ratios
Main effect of Factor A (rows)
Main effect of Factor B (columns)
Interaction effect of Factors A and B

14. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Stage 2
Factor A variance
Factor B variance
Interaction variance

15. Figuring a Two-Way ANOVA
Sums of squares

16. Figuring a Two-Way ANOVA
Degrees of freedom
Number of levels of B
Number of levels of A

17. Figuring a Two-Way ANOVA
Means squares (estimated variances)

18. Figuring a Two-Way ANOVA
F-ratios

19. Factor B: Arousal Level
Low B1
Medium B2
High B3
FactorA:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 11 10 8 A2
Difficult
Example

20. Factor B: Arousal Level
Low B1
Medium B2
High B3
FactorA:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 11 10 8 A2
Difficult
Example

21. Factor B: Arousal Level
Low B1
Medium B2
High B3
FactorA:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 11 10 8 A2
Difficult
Example

22. Factor B: Arousal Level
Low B1
Medium B2
High B3
FactorA:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 11 10 8 A2
Difficult
Example

23. Example: ANOVA table Source SS df MS F Between A B AB 120 60 1 2 30
27.7
6.9
Within
Total
104
344 24
4.33 √ √ √

24. Factorial ANOVA in SPSS
What we covered today is a completely between groups Factorial ANOVA
Enter your observations in one column, use separate columns to code the levels of each factor
Analyze -> General Linear Model -> Univariate
Enter your dependent variable (your observations)
Enter each of your factors (IVs)
Output
Ignore the corrected model, intercept, & total (for now)
F for each main effect and interaction

25. Assumptions in Two-Way ANOVA
Populations follow a normal curve
Populations have equal variances
Assumptions apply to the populations that go with each cell

26. Effect Size in Factorial ANOVA (completely between groups)
Note: if you downloaded the lecture Tues. there were two errors

27. Approximate Sample Size Needed in Each Cell for 80% Power (
Approximate Sample Size Needed in Each Cell for 80% Power (.05 significance level)

28. Other ANOVA designs Basics of repeated measures factorial ANOVA
Using SPSS
Basics of mixed factorial ANOVA
Similar to the between groups factorial ANOVA
Main effects and interactions
Multiple sources for the error terms (different denominators for each main effect)

29. Example
Suppose that you are interested in how sleep deprivation impacts performance. You test 5 people on two tasks (motor and math) over the course of time without sleep (24 hrs, 36 hrs, and 48 hrs). Dependent variable is number of errors in the tasks.
Both factors are manipulated as within subject variables
Need to conduct a within groups factorial ANOVA

30. Example Factor B: Hours awake Factor A: Task 24 36 48 Motor Math B1 B21 4 3 5 6 9 A2
Math 2

31. Within factorial ANOVA in SPSS
Each condition goes in a separate column
It is to your benefit to systematically order those columns to reflect the factor structure
Make your column labels informative
Analyze -> General Linear Model -> Repeated measures
Enter your factor 1 & number of levels, then factor 2 & levels, etc. (remember the order of the columns)
Tell SPSS which columns correspond to which condition
As was the case before, lots of output
Focus on the within-subject effects
Note: each F has a different error term

32. Example Source SS df MS F p A Error (A) 1.20 13.13 1 4 3.28 0.37 0.58 B
Error (B) AB
Error (AB)
104.60
6.10
2.60
8.10 2 8
52.30
0.76
1.30
1.01
< 0.01

33. Example
It has been suggested that pupil size increases during emotional arousal. A researcher presents people with different types of stimuli (designed to elicit different emotions). The researcher examines whether similar effects are demonstrated by men and women.
Type of stimuli was manipulated within subjects
Sex is a between subjects variable
Need to conduct a mixed factorial ANOVA

34. Example Factor B: Stimulus FactorA: Sex Neutral Pleasant Aversive Men4 3 2 8 6 5 1 A2
Women 7

35. Mixed factorial ANOVA in SPSS
Each within condition goes in a separate column
It is to your benefit to systematically order those columns to reflect the factor structure
Make your column labels informative
Each between groups factor has a column that specifies group membership
Analyze -> General Linear Model -> Repeated measures
Enter your within groups factors: factor 1 & number of levels, then factor 2 & levels, etc. (remember the order of the columns)
Tell SPSS which columns correspond to which condition
Enter your between groups column that specifies group membership
As was the case before, lots of output
Need to look at the within-subject effects and the between groups effects

36. Example Source SS df MS F p Between Sex (A) Error (A) 0.83 20.00 1 8
2.50
Within
Stimulus (B)
Sex * Stimulus
Error (B)
58.10
0.07
39.20 2 16
29.00
0.03
2.45

37. Partitioning the variance
Stage 1 partition is same as usual
Stage 2 combines the other partitioning that we’ve done:
The between subjects var is broken into 2 parts
The within subjects is broken into different parts.
Note: the interaction, because it involves a within groups variable, comes out in the partitioning of the within groups par