## Lesson 7: Repeated-Measures ANOVA

#### Objectives

- Conduct the repeated-measures ANOVA.
- Interpret the output.
- Construct a profile plot.

#### Overview

The repeated-measures or within-subjects ANOVA is used when there are multiple measures for each participant. It is conceptually useful to think of the repeated-measures ANOVA as an extension of the paired-samples *t* test. Each set of observations for a subject or case serves as its own control, so this test is quite powerful. In the repeated-measures ANOVA, the test of interest is the within-subjects effect of the treatments or repeated measures.

The procedure for performing a repeated-measures ANOVA in SPSS is found in the **Analyze**, **General Linear Model** menu.

#### Example Data

Assume that a statistics professor is interested in the effects of taking a statistics course on performance on an algebra test. She administers a 20-item college algebra test to ten randomly selected statistics students at the beginning of the term, at the end of the term, and six months after the course is finished. The hypothetical test results are as follows.

Student |
Before |
After |
SixMo |

1 |
13 |
15 |
17 |

2 |
8 |
8 |
7 |

3 |
12 |
15 |
14 |

4 |
12 |
17 |
16 |

5 |
19 |
20 |
20 |

6 |
10 |
15 |
14 |

7 |
10 |
13 |
15 |

8 |
8 |
12 |
11 |

9 |
14 |
15 |
13 |

10 |
11 |
16 |
9 |

#### Coding Considerations

Data coding considerations in the repeated-measures ANOVA are similar to those in the paired-samples *t* test. Each participant or subject takes up a single row in the data file, and each observation requires a separate column. The properly coded SPSS data file with the data entered correctly should appear as follows (see figure 7-1). You may also retrieve a copy of the data file if you like.

Figure 7-1 SPSS data file coded for repeated-measures ANOVA

#### Performing the Repeated-Measures ANOVA

To perform the repeated-measures ANOVA in SPSS, click on **Analyze**, then **General Linear Model**, and then **Repeated Measures**. See Figure 7-2.

Figure 7-2 Select Analyze, General Linear Model, Repeated Measures

In the resulting Repeated Measures dialog, you must specify the number of factors and the number of levels for each factor. In this case, the single factor is the time the algebra test was taken, and there are three levels: at the beginning of the course, immediately after the course, and six months after the course. You can accept the default label of factor1, or change it to a more descriptive one. We will use "Time" as the label for our factor, and specify that there are three levels (see Figure 7-3).

Figure 7-3 Specifying factor and levels

After naming the factor and specifying the number of levels, you must add the factor and then define it. Click on **Add** and then click on **Define**. See Figure 7-4.

Figure 7-4 Specifying within-subjects variable levels

Now you can enter the levels one at a time by clicking on a variable name and then clicking on the right arrow adjacent to the Within-Subjects Variables field. Or you can click on Before in the left pane of the Repeated Measures dialog, then hold down <Shift> and click on SixMo to select all three levels at the same time, and then click on the right arrow to move all three levels to the window in one step (see Figure 7-5).

Figure 7-5 Within-subjects variables appropriately entered

Clicking on **Options** allows you to specify the calculation of descriptive statistics, effect size, and contrasts among the means. If you like, you can also click on **Plots** to include a line graph of the algebra test mean scores for the three administrations. Figure 7-6 is a screen shot of the Profile Plots dialog. You should click on **Time**, then **Horizontal** **Axis**, and then click on **Add**. Click **Continue** to return to the Repeated Measures dialog.

Figure 7-6 Profile Plots dialog

Now click on **Options** and specify descriptive statistics, effect size, and contrasts (see Figure 7-7). You must move Time to the Display Means window as well as specify a confidence level adjustment for the main effects contrasts. A Bonferroni correction will adjust the alpha level in the post hoc comparisons, while the default LSD (Fisher's least significant difference test) will not adjust the alpha level. We will select the more conservative Bonferroni correction.

Figure 7-7 Specifying descriptive statistics, effect size, and mean contrasts

Click on **Continue**, then **OK** to run the repeated-measures ANOVA. The SPSS output provides several tests. When there are multiple dependent variables, the multiviariate test is used to determine whether there is an overall within-subjects effect for the combined depedendent variables. As there is only one within-subject factor, we can ignore this test in the present case. Sphericity is an assumption that the variances of the differences between the pairs of measures are equal. The insignificant test of sphericity indicates that this assumption is not violated in the present case, and adjustments to the degrees of freedom (and thus to the *p* level) are not required. The test of interest is the Test of Within-Subjects Effects. We can assume sphericity and report the *F* ratio as 8.149 with 2 and 18 degrees of freedom and the *p* level as .003 (see Figure 7-8). Partial eta-squared has an interpretation similar to that of eta-squared in the one-way ANOVA, and is directly interpretable as an effect-size index: about 48 percent of the within-subjects variation in algebra test performance can be explained by knowledge of when the test was administered.

Figure 7-8 Test of within-subjects effects

Additional insight is provided by the Bonferroni-corrected pairwise comparisons, which indicate that the means for Before and After are significantly different, while none of the other comparisons are signficant. The profile plot is of assistance in the visualization of these contrasts. See Figures 7-9 and 7-10. These results indicate an immediate but unsustained improvement in algebra test performance for students taking a statistics course.

Figure 7-9 Bonferroni-corrected pairwise comparisions

Figure 7-10 Profile plot