1. Chapter Fourteen The Two-Way Analysis of Variance

2. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 2 New Statistical Notation 1.The two-way ANOVA is the parametric inferential procedure performed when an experiment contains two independent variables 2.When both factors involve independent samples, we perform the two-way, between-subjects ANOVA 3.When both factors involve related samples, we perform the two-way, within-subjects ANOVA 4.When one factor is tested using independent samples and the other factor using related samples, we perform the two-way, mixed-design ANOVA

3. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 3 Understanding the Two-Way ANOVA

4. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 4 Factorial Designs When we combine all levels of one factor with all levels of the other factor, this produces a complete factorial design When all levels of the two factors are not combined, this produces an incomplete factorial design

5. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 5 Overview of the Two-Way Between-Subjects ANOVA

6. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 6 Assumptions of the Two-Way Between-Subjects ANOVA 1.Each cell contains an independent sample 2.The dependent variable measures interval or ratio scores that are approximately normally distributed 3.The populations have homogenous variance

7. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 7 Main Effects The main effect of a factor is the effect that changing the levels of that factor has on dependent variable scores while ignoring all other factors in the study We collapse across a factor. Collapsing across a factor means averaging together all scores from all levels of that factor.

8. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 8 Interaction Effects The interaction of two factors is called a two- way interaction The two-way interaction effect is the influence on scores that results from combining the levels of factor A with the levels of factor B When you look for the interaction effect, you compare the cell means. When you look for a main effect, you compare the level means.

9. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 9 Interaction Effect An interaction effect is present when the relationship between one factor and the dependent scores change with, or depends on, the level of the other factor that is present A two-way interaction effect indicates that the influence that one factor has on scores depends on which level of the other factor is present

10. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 10 Summary Table of a Two-way ANOVA

11. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 11 Computing the Two-Way ANOVA

12. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 12 Computing F obt 1.Compute the total sum of squares (SS tot )

13. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 13 Computing F obt 2.Compute the sum of squares between groups for column factor A (SS A )

14. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 14 Computing F obt 3.Compute the sum of squares between groups for row factor B (SS B )

15. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 15 Computing F obt 4.Compute the overall sum of squares between groups (SS bn )

16. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 16 Computing F obt 5.Compute the sum of squares between groups for the interaction (SS A x B )

17. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 17 Computing F obt 6.Compute the sum of squares within groups (SS wn )

18. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 18 Computing F obt 7.Compute the degrees of freedom 1.The degrees of freedom between groups for factor A is k A - 1 2.The degrees of freedom between groups for factor B is k B - 1 3.The degrees of freedom between groups for the interaction is (df A )(df B ) 4.The degrees of freedom within groups equals N – k AxB 5.The degrees of freedom total equals N - 1

19. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 19 8.Compute the mean squares 1. 2. 3. Computing F obt 4.4.

20. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 20 9.Compute F obt 1. 2. 3. Computing F obt

21. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 21 Interpreting the Two-Way Experiment

22. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 22 Graphing the Effects To graph main effects, plot the dependent variable along the Y axis and the levels of a factor along the X axis To graph interaction effects, plot the dependent variable along the Y axis. Place the levels of one factor along the X axis, and show the second factor by drawing a separate line connecting the means for each level of that factor.

23. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 23 Graphs Showing Main Effects

24. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 24 Graph of Cell Means, Showing the Interaction

25. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 25 Two Graphs Showing When an Interaction Is and Is Not Present

26. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 26 Performing Post Hoc Comparisons Perform post hoc comparisons on the level means from significant main effect using Tukey’s HSD Perform Tukey’s HSD for the interaction using only unconfounded comparisons –A confounded comparison occurs when two cells differ along more than one factor –An unconfounded comparison occurs when two cells differ along only one factor

27. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 27 Describing the Effect Size Compute eta squared to describe effect size. That is, the proportion of variance in dependent scores that is accounted for by a manipulation.

28. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 28 Confidence Interval The computational formula for the confidence interval for a single  is

29. Factor A Group A1 Group A2 Group A3 Factor B Group B1 14 10131115 131012111413 Group B2 171810121412 191611101415 Example Using the following data set, conduct a two-way ANOVA. Use  = 0.05

30. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 30 Example

31. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 31 Example

32. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 32 Example

33. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 33 Example

34. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 34 Example df A = 3 - 1 = 2 df B = 2 - 1 = 1 df A X B = (2)(1) = 2 df wn = 24 - 6 = 18

35. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 35 Example

36. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 36 Example

37. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 14 - 37 Example F obt for 2 and 18 degrees of freedom is 3.55 F obt for 1 and 18 degrees of freedom is 4.41 The main effect for Factor A is significant The main effect for Factor B is significant The interaction term (A X B) is significant