So far, we have worked with univariate data (one variable measured across two samples), and we wanted to test whether the means are equal or not. In certain cases, we might work with multivariate data (for example, measurements of height and weight for certain individuals), and we will be interested in testing the multivariate hypothesis, which is that the means for all of the variables are equal between two groups or not. This is usually formulated as follows:
The difference is that each element is a vector, and we are testing whether all of the elements in a vector are the same between groups. The main assumption here (similar to the univariate t-test) is that the data comes from a multivariate Gaussian distribution.
A relevant question at this stage is whether we can ignore the multi-dimensionality of the problem, and just do univariate t-tests. This would...