## Lesson 8: Two-Way ANOVA

#### Objectives

1. Conduct the two-way ANOVA.
2. Examine and interpret main effects and interaction effect.
3. Produce a plot of cell means.

#### Overview

We will introduce the two-way ANOVA with the simplest of such designs, a balanced or completely-crossed factorial design. In this case there are two independent variables (factors), each of which has two or more levels. We can think of this design as a table in which each cell represents a single independent group. The group represents a combination of levels of the two factors. For simplicity, let us refer to the factors as A and B and assume that each factor has two levels and each independent group has the same number of observations. There will be four independent groups. The design can thus be visualized as follows: Figure 8-1 Conceptualization of Two-Way ANOVA

The two-way ANOVA is an economical design, because it allows the assessment of the main effects of each factor as well as their potential interaction.

#### Example Data and Coding Considerations

Assume that you are studying the effects of observing violent acts on subsequent aggressive behavior. You are interested in the kind of violence observed: a violent cartoon versus a video of real-action violence. A second factor is the amount of time one is exposed to violence: ten minutes or 30 minutes. You randomly assign 8 children to each group. After the child watches the violent cartoon or action video, the child plays a Tetris-like computer video game for 30 minutes. The game provides options for either aggressing ("trashing" the other computerized player) or simply playing for points without interfering with the other player. The program provides 100 opportunities for the player to make an aggressive choice and records the number of times the child chooses an aggressive action when the game provides the choice. The hypothetical data are below: Figure 8-2 Example Data

When coding and entering data for this two-way ANOVA, you should recognize that each of the 32 participants is a unique individual and that there are no repeated measures. Therefore, each participant takes up a row in the data file, and the data should be coded and entered in such a way that the factors are identified by two columns with group membership coded as a combination of the levels. For illustrative purposes we will use 1 and 2 to represent the levels of the factors, though as you learned earlier, you could just as easily have used 0s and 1s. The data view of the resulting SPSS data file should appear something like this: Figure 8-3 SPSS data file data view for two-way ANOVA (partial data)

For ease of interpretation, the variables can be labeled and the values of each specified in the variable view (see Figure 8-4). Figure 8-4 Variable view with labels and values identified

If you prefer, you may retrieve a copy of the data file.

#### Performing the Two-Way ANOVA

To perform the two-way ANOVA, select Analyze, General Linear Model, and then Univariate because there is only one dependent variable (see Figure 8-5). Figure 8-5 Select Analyze, General Linear Model, Univariate

In the resulting dialog, you should specify that Aggression is the dependent variable and that both Time and Type are fixed factors (see Figure 8-6). Figure 8-6 Specifying the two-way ANOVA

This procedure will test the main effects for Time and Type as well as their possible interaction. It is helpful to specify profile plots to examine the interaction of the two variables. For that purpose, select Plots and then move Type to the Horizontal Axis field and Time to the Separate Lines field (see Figure 8-7). Figure 8-7 Specifying profile plots

When you click on Add, the Type * Time interaction is added to the Plots window, as shown in Figure 8-8. Figure 8-8 Plotting an interaction term

Click Continue, then click Options. Check the boxes in front of Descriptive statistics and Estimates of effect size (see Figure 8-9). Click Continue, then click OK to run the two-way ANOVA. The table of interest is the Test of Between-Subjects Effects. Examination of the table reveals significant F ratios for Time, Type and the Time * Type interaction (see Figure 8-9). Figure 8-9 Table of between-subjects effects

As in the repeated-measures ANOVA, a partial eta-squared is calculated as a measure of effect size. The profile plot (see Figure 8-10) shows that the interaction is ordinal: the differences in the number of aggressive choices made after observing the two violence conditions increase with the time of exposure. Figure 8-10 Interaction plot