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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 • How to perform the Tukey HSD test on the interaction • How to interpret the results of a two-way experiment Understanding the Two-Way Design The Two-Way ANOVA 1.The two-way ANOVA is the parametric inferential procedure performed when a design involves two independent variables 2.When both factors involve independent samples, we perform the two-way betweensubjects ANOVA The Two-Way ANOVA 3. When both factors involve related measures, we perform the two-way within-subjects ANOVA 4. When one factor is tested using independent samples and the other factor involves related samples, we perform the two-way mixed-design ANOVA Organization 1. Each column represents a level of factor A 2. Each row represents a level of factor B 3. Each square represents combining a level of factor A with a level of factor B and is called a cell 4. When we combine all levels of one factor with all levels of the other factor, the design is called a factorial design Organization Understanding the Main Effects 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. Main Effects Means The mean of the level of one factor after collapsing the other factor is known as the main effect mean. Statistical Hypotheses • Factor A H 0 : A1 A2 A3 Ha: At least two of the main effect means are different • Factor B H 0 : B1 B2 Ha: At least two of the main effect means are different Understanding the Interaction Effect Interaction Effects The interaction of two factors is called a twoway 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. 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 Completing the Two-Way Design Summary Table 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 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. 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 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 Graphing the Interaction Interpreting the Two-Way Experiment 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 Nonsignificant Interaction When the interaction is not significant, interpretation of the main effects can occur. Main Effect and Interaction Means Example Use the following data to conduct a two-way ANOVA Group 1 Group 2 Group 3 4th Graders 14 14 10 10 13 12 11 15 14 5th Graders 14 14 15 11 11 10 13 14 15 Example Summary Table Sum of Squares Source df Mean Square F Between Factor A (Group) 23.444 2 11.722 5.146 Factor B (Grade) 0.889 1 0.889 0.390 Interaction 5.444 2 2.722 1.195 Within 27.333 12 2.278 Total 57.111 17 * Indicates significant at a = 0.05 * Example Since Factor A (Group) is significant… k = 3 and n = 6, dfwn = 12, and a = .05, so q = 3.77 MS wn HSD (qk ) n 2.278 (3.77) 6 2.323 Example X Group1 13.50 X Group 2 11.17 X Group3 13.67 • 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 Example Effect size for Factor A (Group) SS bn 23.444 h 0.4105 SStot 57.111 2 Example Graphing the means Group 16 14 12 10 8 6 4 2 0 1 2 3