We have already explained that ANOVA is the generalization of the t-test for multiple samples. On the other hand, the t-test is designed to work with just one variable, and in case we have multiple variables, we need to use Hotelling T2. Is it possible to extend ANOVA to work with multiple variables? The answer is yes, and the technique for doing so is called MANOVA (Multiple ANOVA). The assumptions for MANOVA are similar to the ones we have when using Hotelling T2 for two samples: equality of covariance matrices between the groups, the data should come from a multivariate Gaussian distribution for each group, and there should not be outliers.

In this example, we will generate data from a multivariate Gaussian distribution (with three variables) with an arbitrary covariance matrix. We will assign the columns' names to represent the History, Math, and Biology scores for three classes (in each class, we will have 50 students). Of course, in real exercises, we will receive our data and we won't be generating any data ourselves. The reason we choose to work this simulated dataset here is because we want to do our MANOVA tests in a controlled environment where we know the true parameters.