1. Statistics for the Social Sciences
Psychology 340
Fall 2013
Thursday, October 24
Factorial Analysis of Variance (ANOVA)

2. Homework #9 (handout) due10/29 Homework #10 due10/31
Ch 14 # 1,2,4,5,7,9, SKIP PROBLEM 10, 14,15, 22 (DO NOT use SPSS for #22)

3. Last Time Brief review of Repeated Measures ANOVA
Assumptions in Repeated Measures ANOVA
Effect sizes in Repeated Measures ANOVA
More practice with SPSS

4. This Time 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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 √

13. 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 √

14. 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

15. 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

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

17. Figuring a Two-Way ANOVA
Sums of squares

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

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

20. Figuring a Two-Way ANOVA
F-ratios

21. Figuring a Two-Way ANOVA
ANOVA table for two-way ANOVA

22. Factor B: Arousal Level
Low B1
Medium B2
High B3
Factor A:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult

23. Factor B: Arousal Level
Low B1
Medium B2
High B3
FactorA:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult

24. Factor B: Arousal Level
Low B1
Medium B2
High B3
Factor A:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult

25. Factor B: Arousal Level
Low B1
Medium B2
High B3
Factor A:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult

26. Example Source SS df MS F Between A B AB 120 60 1 2 30 24.0 6.0 Within
Total
360 24 5 √ √ √

27. Computational Formulas
T = Group (Condition) Total
G = Grand Total
TRow = Row Total
Tcolumn=Column Total
n = number of participants in treatment
nrow=number of participants in row
ncolumn=number participants in col.
N = number of participants
Same as in one-way ANOVA
Same as in one-way ANOVA, but T refers to each treatment or condition (e.g., A1B1 is one treatment)
Same as in one-way ANOVA, but each treatment or condition is one cell in the design (e.g., A1B1 is one treatment)

28. Factor B: Arousal Level
Low B1
Medium B2
High B3
Factor A:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult
TROW1= 90
TROW2= 30
TCOLUMN1= 20
TCOLUMN2= 50
TCOLUMN1= 50
G=120

29. Computational Formulas
T = Group (Condition) Total
G = Grand Total
TRow = Row Total
Tcolumn=Column Total
n = number of participants in treatment
nrow=number of participants in row
ncolumn=number participants in col.
N = number of participants

30. Factor B: Arousal Level
Low B1
Medium B2
High B3
Factor A:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult
Example

31. Computational Formulas
T = Group (Condition) Total
G = Grand Total
TRow = Row Total
Tcolumn=Column Total
n = number of participants in treatment
nrow=number of participants in row
ncolumn=number participants in col.
N = number of participants

32. Factor B: Arousal Level
Low B1
Medium B2
High B3
FactorA:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult
TROW1= 90
TROW2= 30
TCOLUMN1= 20
TCOLUMN2= 50
TCOLUMN1= 50
G=120

33. Computational Formulas
T = Group (Condition) Total
G = Grand Total
TRow = Row Total
Tcolumn=Column Total
n = number of participants in treatment
nrow=number of participants in row
ncolumn=number participants in col.
N = number of participants

34. Factor B: Arousal Level
Low B1
Medium B2
High B3
FactorA:
Task
Difficulty A1
Easy 3 1 6 4 2 5 9 7 13 8 A2
Difficult

35. Effect Size in Factorial ANOVA

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

37. Extensions and Special Cases of the Factorial ANOVA
Three-way and higher ANOVA designs
Repeated measures ANOVA

38. Factorial ANOVA in Research Articles
A two-factor ANOVA yielded a significant main effect of voice, F(2, 245) = 26.30, p < As expected, participants responded less favorably in the low voice condition (M = 2.93) than in the high voice condition (M = 3.58). The mean rating in the control condition (M = 3.34) fell between these two extremes. Of greater importance, the interaction between culture and voice was also significant, F(2, 245) = 4.11, p < .02.

39. Factorial ANOVA in SPSS
Analyze=>General Linear Model=>Univariate
Highlight the column label for the dependent variable in the left box and click on the arrow to move it into the Dependent Variable box.
One by one, highlight the column labels for the two factor codes (Independent Variables) and click the arrow to move them into the Fixed Factors box.
If you want descriptive statistics for each treatment, click on the Options box, select Descriptives, and click continue.
Click OK
In the output, look at “corrected model” and ignore “intercept” and look at “corrected total” and ignore “total.”