## Lesson 9: ANOVA for Mixed Factorial Designs

#### Objectives

- Conduct a mixed-factorial ANOVA.
- Test between-groups and within-subjects effects.
- Construct a profile plot.

#### Overview

A mixed factorial design involves two or more independent variables, of which at least one is a within-subjects (repeated measures) factor and at least one is a between-groups factor. In the simplest case, there will be one between-groups factor and one within-subjects factor. The between-groups factor would need to be coded in a single column as with the independent-samples *t* test or the one-way ANOVA, while the repeated measures variable would comprise as many columns as there are measures as in the paired-samples *t* test or the repeated-measures ANOVA.

#### Example Data

As an example, assume that you conducted an experiment in which you were interested in the extent to which visual distraction affects younger and older people's learning and remembering. To do this, you obtained a group of younger adults and a separate group of older adults and had them learn under three conditions (eyes closed, eyes open looking at a blank field, eyes open looking at a distracting field of pictures). This is a 2 (age) x 3 (distraction condition) mixed factorial design. The scores on the data sheet below represent the number of words recalled out of ten under each distraction condition.

Age |
Closed Eyes |
Simple Distraction |
Complex Distraction |

Younger |
8 |
5 |
3 |

Younger |
7 |
6 |
6 |

Younger |
8 |
7 |
6 |

Younger |
7 |
5 |
4 |

Older |
6 |
5 |
2 |

Older |
5 |
5 |
4 |

Older |
5 |
4 |
3 |

Older |
6 |
3 |
2 |

#### Building the SPSS Data File

Note that there are eight separate participants, so the data file will require eight rows. There will be a column for the participants' age, which is the between-groups variable, and three columns for the repeated measures, which are the distraction conditions. As always it is helpful to include a column for participant (or case) number.

The data appropriately entered in SPSS should look something like the following (see Figure 9-1). You may optionally download a copy of the data file.

Figure 9-1 SPSS data structure for mixed factorial design

#### Performing the Mixed Factorial Anova

To conduct this analysis, you will use the repeated measures procedure. The initial steps are identical to those in the within-subjects ANOVA. You must first specify repeated measures to identify the within-subjects variable(s), and then specify the between-groups factor(s).

Select **Analyze**, then **General Linear Model**, then **Repeated Measures** (see Figure 9-2).

.

Figure 9-2 Preparing for the Mixed Factorial Analysis

Next, you must define the within-subjects factor(s). This process should be repeated for each factor on which there are repeated measures. In our present case, there is only one within-subject variable, the distraction condition. SPSS will give the within-subjects variables the names factor1, factor2, and so on, but you can provide more descriptive names if you like. In the Repeated Measures dialog box, type in the label distraction and the number of levels, 3. See Figure 9-3. If you like, you can give this measure (the three distraction levels) a new name by clicking in the Measure Name field. If you choose to name this factor, the name must be unique and may not conflict with any other variable names. If you do not name the measure, the SPSS name for the measure will default to MEASURE_1. In the present case we will leave the measure name blank and accept the default label.

Figure 9-3 Specifying the within-subjects factor.

We will now specify the within-subjects and between-groups variables. Click on **Add** and then **Define** to specify which variable in the dataset is associated with each level of the within-subjects factor (see Figure 9-4).

Figure 9-4 Defining the within-subjects variable

Move the Closed, Simple, and Complex variables to levels 1, 2, and 3, respectively, and then move Age to the Between-Subjects Factor(s) window (see Figure 9-5). You can optionally specify one or more covariates for analysis of covariance.

Figure 9-5 The complete design specification for the mixed factorial ANOVA

To display a plot of the cell means, click on **Plots**, and then move Age to the Horizontal axis, and distraction to Separate Lines. Next click on **Add** to specify the plot (see Figure 9-6) and then click **Continue**.

Figure 9-6 Specifying plot

We will use the **Options** menu to specify the display marginal and cell means, to compare main effects, to display descriptive statistics, and display measures of effect size. We will select the Bonferroni interval adjustment to control the level of Type I error. See Figure 9-7.

Figure 9-7 Repeated measures options

Select **Continue** to close the options dialog and then **OK** to run the ANOVA. The resulting SPSS output is rather daunting, but you should focus on the between and within-subjects tests. The test of sphericity is not significant, indicating that this assumption has not been violated. Therefore you should use the *F* ratio and degrees of freedom associated with the sphericity assumption (see Figure 9-8). Specifically you will want to determine whether there is a main effect for age, an effect for distraction condition, and a possible interaction of the two. The tables of interest from the SPSS Viewer are shown in Figures 9-8 and 9-9.

Figure 9-8 Partial SPSS output

The test of within-subjects effects indicates that there is a significant effect of the distraction condition on word memorization. The lack of an interaction between distraction and age indicates that this effect is consistent for both younger and older participants. The test of between-subjects effects (see Figure 9-9) indicates there is a significant effect of the age condition on word memory.

Figure 9-9 Test of between-subjects effects

The remainder of the output assists in the interpretation of the main effects of the within-subjects (distraction condition) and between-subjects (age condition) factors. Of particular interest is the profile plot, which clearly displays the main effects and the absence of an interaction (see Figure 9-10). As disussed above, SPSS calls the within subjects variable MEASURE_1 in the plot.

Figure 9-10 Profile plot