Multivariate Designs - Between-Groups Designs
All examples discussed so far have involved only one dependent variable. Even though the computations become increasingly complex, the logic and nature of the computations do not change when there is more than one dependent variable at a time.
For example, we may conduct a study where we try two different textbooks, and we are interested in the students' improvements in math and physics. In that case, we have two dependent variables, and our hypothesis is that both together are affected by the difference in textbooks. We could now perform a multivariate analysis of variance (MANOVA) to test this hypothesis. Instead of a univariate F value, we would obtain a multivariate F value (Wilks Lambda) based on a comparison of the error variance/covariance matrix and the effect variance/covariance matrix. The "covariance" here is included because the two measures are probably correlated and we must take this correlation into account when performing the significance test. Obviously, if we were to take the same measure twice, then we would really not learn anything new. If we take a correlated measure, we gain some new information, but the new variable will also contain redundant information that is expressed in the covariance between the variables.
Interpreting results
If the overall multivariate test is significant, we conclude that the respective effect (e.g., textbook) is significant. However, our next question would of course be whether only math skills improved, only physics skills improved, or both. In fact, after obtaining a significant multivariate test for a particular main effect or interaction, customarily one would examine the univariate F-tests for each variable to interpret the respective effect. In other words, one would identify the specific dependent variables that contributed to the significant overall effect.