One common use of multivariate fitting is to analyze data with repeated measures, also called longitudinal data. A subject is measured repeatedly across time, and the data are arranged so that each of the time measurements form a variable. Because of correlation between the measurements, data should not be stacked into a single column and analyzed as a univariate model unless the correlations form a pattern termed sphericity. See the previous section, Univariate Tests and the Test for Sphericity, for more information about this topic.
With repeated measures, the analysis is divided into two layers:
• Between-subject (or across-subject) effects are modeled by fitting the sum of the repeated measures columns to the model effects. This corresponds to using the Sum response function. This response function is an M-matrix that is a single vector of 1s.
• Within-subjects effects (repeated effects, or time effects) are modeled with a response function that fits differences in the repeated measures columns. This analysis can be done using the Contrast response function or any of the other similar differencing functions: Polynomial, Helmert, Profile, or Mean. When you model differences across the repeated measures, think of the differences as being a new within-subjects effect, usually time. When you fit effects in the model, interpret them as the interaction with the within-subjects effect. For example, the effect for Intercept becomes the Time (within-subject) effect, showing overall differences across the repeated measures. If you have an effect A, the within-subjects tests are interpreted to be the tests for the A*Time interaction, which model how the differences across repeated measures vary across the A effect.
Figure 9.1 shows the relationship between the response function and the model effects compared with what a univariate model specification would be. Using both the Sum (between-subjects) and Contrast (within-subjects) models, you should be able to reconstruct the tests that would have resulted from stacking the responses into a single column and obtaining a standard univariate fit.
There is a direct and an indirect way to perform the repeated measures analyses:
• The direct way is to use the pop-up menu item Repeated Measures. This prompts you to name the effect that represents the within-subject effect across the repeated measures. Then it fits both the Contrast and the Sum response functions. An advantage of this way is that the effects are labeled appropriately with the within-subjects effect name.
• The indirect way is to specify the two response functions individually. First, do the Sum response function and second, do either Contrast or one of the other functions that model differences. You need to remember to associate the within-subjects effect with the model effects in the contrast fit.