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Chapter 12

Chapter 12. Understanding the Two-Way Analysis of Variance. 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. Going Forward.

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Chapter 12

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  1. 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 effectmeans 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 Ha: At least two of the main effect means are different

  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 dependson 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

  28. Example Summary Table * Indicates significant at a = 0.05

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

  30. Example • Since 13.5 – 11.17 = 2.33 is greater than 2.323, the mean for Group 1 is significantly different from the mean for Group 2 • Likewise, since 13.67 – 11.17 = 2.50 is greater than 2.323, the mean for Group 3 is significantly different from the mean for Group 2 • But 13.67 – 13.50 = 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

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