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# Repeated measures with multiple dependent variables

### This is an excerpt from Research Methods in Physical Activity-8th Edition by Jerry R. Thomas,Philip E. Martin,Jennifer L. Etnier & Stephen J. Silverman.

Rather frequently, experiments have multiple dependent variables that are measured more than once (over time). For example, in an exercise adherence study, both physiological and psychological variables might be measured once a week during a 15-week training program. If there were two training groups (different levels of training) and a control group, each with 15 participants, all of whom were measured every week (i.e., 15 times) on two psychological and three physiological measures, we would have a design that is three levels of exercise ×15 trials (3 × 15) for five dependent variables. This design offers several options for analysis.

We could do five 3 × 15 ANOVAs with repeated measures on the second factor. Here, we would follow the repeated-measures procedures described earlier in this chapter. But we would be inflating the alpha by doing multiple analyses on the same subjects. Of course, the alpha could be adjusted by the Bonferroni technique (𝛼 = .05⁄5 = .01), but this fails to consider the potentially substantial and possibly very interesting interrelationships of the dependent variables. Schutz and Gessaroli (1987) provide an excellent tutorial on how to handle this problem. The following is a brief discussion of their study, but if you are using this design and analysis, you should read their complete study and example.

Two options are available for this analysis: multivariate mixed-model (MMM) analysis and doubly multivariate (DM) analysis. Which is used depends on the assumptions that your data meet. Using the study previously described, three levels of exercise ×15 trials for five dependent variables, the MMM analysis treats the independent variable (levels of exercise) as a true multivariate case by forming a linear composite of the five dependent variables to discriminate between the levels of the independent variable. If this composite is significant, it can be followed up by the stepdown F procedure (for an alternative procedure, see Schutz & Gessaroli, 1987). For the repeated-measures factor (and interaction), a linear composite is formed and treated as a regular repeated-measures analysis for each trial. This means that the sphericity assumption must be met as previously described, and the epsilon can be used to test this assumption with the same standards previously described. The interpretation of the resulting F ratio for main effects for groups, trials, and the group × trial interaction are the same as in other designs. The question usually posed here is Do the groups change at different rates across the trials on the linear composite of dependent variables? If the sphericity assumption can be met, MMM analysis is preferred because most authors believe that it offers more power. This assumption is difficult to meet, however, especially if there are more than two dependent variables measured on more than three trials.

The DM analysis does not require that the sphericity assumption be met. The analysis is the same for the independent variable of exercise. In the repeated-measures part of the analysis, however, a linear composite is formed not only of the dependent variables at each trial but also of the 15 trials, which themselves become a linear composite; thus the name doubly multivariate. The interpretations of the Fs for the two main effects and interaction remain essentially the same, but follow-ups become more complex.

McCullagh and Meyer (1997) compared four methods of providing information (physical practice with feedback, learning model with model feedback, correct model with model feedback, and learning model without model feedback) on learning correct form in the free-weight squat lift. There were two dependent variables (outcome and form) and five trials. The authors analyzed the data using MANOVA with repeated measures. Univariate ANOVAs and post hoc comparisons were done as follow-ups for significant Fs.

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