Specifying Within-Subjects (Repeated Measures) Univariate and Multivariate Designs - One-Way Within-Subjects ANOVA

Often, you want to administer the same test to the same subjects repeatedly over a period of time or under different circumstances. In essence, you are interested in examining differences within each subject, for example, subjects' improvement over time. Such designs are referred to as within-subjects designs or repeated measures designs.

For example, imagine that you want to monitor the improvement of students' algebra skills over three months of instruction. A standardized algebra test is administered after one month (level 1 of the repeated measures factor), and comparable tests are administered after two months (level 2 of the repeated measures factor) and after three months (level 3 of the repeated measures factor). Thus, the repeated measures factor (Time) has three levels.

Setting up the datafile
In order to analyze such a dataset, you need to perform a within-subjects (or repeated measures) analysis of variance. First, a data file should be created that contains the test scores for all subjects; this file should be arranged as follows:
 Test Administered After
 1 Month 2 Months 3 Months
Case 1 24 26 28
Case 2 30 30 32
Case 3 30 29 28
Case 4 30 29 34
Case 5 35 36 32
- - - -
- - - -
- - - -

The first variable (column) contains subjects' scores on the algebra test after one month of instruction, the second variable (column 2) contains the respective subjects' scores on the algebra test after two months of instruction, and variable 3 (column 3) contains the respective subjects' scores after three months of instruction.

Specifying the design
In order to analyze this dataset, first select Repeated measures ANOVA as the Type of analysis and Quick specs dialog as the Specification method on the General ANOVA/MANOVA Startup Panel - Quick tab or GLM Startup Panel Quick tab. Next, click the OK button on the General ANOVA/MANOVA (Startup Panel) or GLM Startup Panel. Then, on the ANOVA/MANOVA Quick Specs - Quick tab or GLM Quick Specs Dialog - Quick tab, click the Variables button and specify Variables 1 to 3 in the Dependent variable list field (leave the categorical predictors list empty). Next, click the Within effects button to specify the repeated measures (within effects). Finally, on the ANOVA Specify Within-Subjects Factor or GLM Specify Within-Subjects Factors dialog, specify one repeated measures factor with three levels (the default in this case).
Testing effects
Effects involving repeated measures factors are tested in exactly the same manner as in between-groups ANOVA. If a repeated measures factor has more than two levels, then there are two alternative ways of assessing the significance of effects involving that factor. The traditional way (see Winer, 1962, 1971) is to perform a univariate test; however, in recent years it has become common practice to use multivariate analysis of variance to analyze such designs. The advantage of the latter approach is discussed in the Introductory Overview; in short, the multivariate approach requires less restrictive assumptions. In practice, the two approaches usually yield similar results unless the changes (differences) across the levels are correlated with each other across subjects. In our example, this would be the case if students' improvement from 1 to 2 months is correlated with their improvement from 2 to 3 months. In any event, it is prudent to carefully evaluate the univariate and multivariate solutions.
Testing planned comparisons
You can test planned comparisons between particular levels of the repeated measures factor in exactly the same manner as you would test planned comparisons in between-groups designs. After clicking the Contrasts for LS means button on the GLM and ANOVA Results - Comps tab, you are asked to enter a set of contrast coefficients.

The rules for generating contrast coefficients are as follows.

  1. The contrast must have as many coefficients as there are levels for the respective factor.
  2. Levels that are to be omitted in the contrast are assigned a 0 (zero).
  3. Levels that are to be compared against each other are assigned positive or negative integer values; however, it is important that the sum of such contrasts is equal to zero.
  4. Levels that are to be collapsed are assigned identical integer values.

Referring back to the example, if you want to compare the students' performance on the algebra test after one month of instruction with their performance after three months of instructions, the following contrast coefficients would be appropriate:

  1 0 -1

If you want to compare subjects' performance after three months of instruction with their performance after one and two months of instruction combined, you would enter the coefficients:

  -1 -1 2