In the multivariate significance test there are the hypotheses of Wilk, Pillai, Hotellng and Roy. Which is the statistical significance of each one? What should I use for my data?
This is a complex topic, there is a huge amount of literature available. And the answer depends on many details of your specific aims and your particular data. Instead of writing book-chapters here explaining all these things once more that are explained in so many courses and books, I suggest that you consult a statistician.