1. Understanding the Two-Way Analysis of Variance
Chapter 12
Understanding the Two-Way Analysis of Variance

2. Going Forward Your goals in this chapter are to learn:
What a two-way ANOVA is
How to calculate main effect means and cell means
What a significant main effect indicates
What a significant interaction indicates

3. Going Forward How to perform the Tukey HSD test on the interaction
How to interpret the results of a two-way experiment

4. Understanding the Two-Way Design

5. The Two-Way ANOVA
The two-way ANOVA is the parametric inferential procedure performed when a design involves two independent variables
When both factors involve independent samples, we perform the two-way between-subjects ANOVA

6. The Two-Way ANOVA
When both factors involve related measures, we perform the two-way within-subjects ANOVA
When one factor is tested using independent samples and the other factor involves related samples, we perform the two-way mixed-design ANOVA

7. Organization Each column represents a level of factor A
Each row represents a level of factor B
Each square represents combining a level of factor A with a level of factor B and is called a cell
When we combine all levels of one factor with all levels of the other factor, the design is called a factorial design

8. Organization

9. Understanding the Main Effects

10. Main Effects
The main effect of a factor is the overall effect changing the levels of that factor has on dependent scores while we ignore the other factor in the study.
To compute a main effect of one factor, we collapse across the other factor. Collapsing a factor refers to averaging together all scores from all levels of that factor.

11. Main Effects Means
The mean of the level of one factor after collapsing the other factor is known as the main effect mean.

12. Statistical Hypotheses
Factor A
Ha: At least two of the main effect means are different
Factor B

13. Understanding the Interaction Effect

14. Interaction Effects
The interaction of two factors is called a two-way interaction
The two-way interaction effect is when the relationship between one factor and the dependent variable changes as the levels of the other factor change
When you look for the interaction effect, you compare the cell means. A cell mean is the mean of the scores from one cell in a two-way design.

15. Interaction Effect
An interaction effect is present when the relationship between one factor and the dependent scores depends on the level of the other factor present
A two-way interaction effect is not present if the cell means form the same pattern regardless of the level of the other factor present

16. Completing the Two-Way Design

17. Summary Table

18. Examining Main Effects
Each main effect is approached as a separate one-way ANOVA
A significant Fobt indicates we should
conduct a Tukey’s HSD test,
compute the effect size, and
graph the means

19. Examining the Interaction
When the interaction effect is significant, we need to
Calculate the effect size using h2,
Graph the interaction,
Conduct a Tukey’s HSD test using only unconfounded comparisons.

20. Examining the Interaction
An unconfounded comparison is one in which two cells differ along only one factor
When two cells differ along both factors, we have a confounded comparison

21. Graphing the Interaction
When graphing the interaction effect,
Label the means of the dependent variable on the Y axis,
Label the factor with the most levels on the X axis, and
Plot the cell means of the factor with fewer levels
A separate line is plotted for each level

22. Graphing the Interaction

23. Interpreting the Two-Way Experiment

24. Significant Interaction
Conclusions about main effects may be contradicted by the interaction
The primary interpretation of a two-way ANOVA may focus on the significant interaction
If the interaction is significant, we do not make conclusions about the main results of the main effects

25. Nonsignificant Interaction
When the interaction is not significant, interpretation of the main effects can occur.

26. Main Effect and Interaction Means

27. Example Use the following data to conduct a two-way ANOVA Group 1
4th Graders 14 10 13 12 11 15
5th Graders

28. * Indicates significant at a = 0.05
Example
Summary Table
* Indicates significant at a = 0.05
Source
Sum of Squares df
Mean Square F
Between
Factor A (Group)
23.444 2
11.722
5.146 *
Factor B (Grade)
0.889 1
0.390
Interaction
5.444
2.722
1.195
Within
27.333 12
2.278
Total
57.111 17

29. Example
Since Factor A (Group) is significant… k = 3 and n = 6, dfwn = 12, and a = .05, so q = 3.77

30. Example
Since 13.5 – = 2.33 is greater than 2.323, the mean for Group 1 is significantly different from the mean for Group 2
Likewise, since – = 2.50 is greater than 2.323, the mean for Group 3 is significantly different from the mean for Group 2
But – = 0.17 is not greater than 2.323, so the mean for Group 1 is not significantly different from the mean for Group 3

31. Example
Effect size for Factor A (Group)

32. Example
Graphing the means