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

• How to perform the Tukey HSD test on the interaction

• How to interpret the results of a two-way experiment

Going Forward

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 between-subjects 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 A2. Each row represents a level of factor B3. 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

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

• Factor B

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

321:0 AAAH

21:0 BBH

Understanding the Interaction Effect

Interaction Effects

The interaction of two factors is called a two-way interactionThe 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 2,•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

* Indicates significant at = 0.05

SourceSum of Squares 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

Example

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

323.2

6

278.2)77.3(

)(

n

MSqHSD wnk

Example67.1317.1150.13 321 GroupGroupGroup XXX

• 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)

4105.0111.57

444.232 tot

bn

SS

SS

Example

Graphing the means