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The pooled matrix is the one that you need to invert, so that is the one that you should examine for high correlations. However, what if you have 2 variables that are highly correlated with each other and several other variables that are fairly independent of those 2 and each other. And what if the two groups are identical on all of the variables other ...

I imagine that productive generalizations would come out from observations that some of these tests are norms of the vector ${\rm spec}[HE^{-1}]=\{\lambda_1, \ldots, \lambda_p\}$, so Hotelling-Lawley's trace is the $l_1$ norm, $\| \{\lambda_1, \ldots, \lambda_p\} \|_1$, and Roy's largest root is the $l_\infty$ norm, $\| \{\lambda_1, \ldots, \lambda_p\} ...

My question was about the distributions, not the test, and I think I've figured out the answer: a t-distribution squared has an F(1,n) distribution, which is a Hotelling distribution (up to rescaling by a constant determined by the parameters). I believe one can say that an F(m,n) distribution is the same as a Wilks' $\Lambda(1,m,n)$ distribution, which is ...

These NCSU course notes say Multivariate tests in contrast to the overall F test, answer the question, "Is each effect significant?" or more specifically, "Is each effect significant for at least one of the dependent variables?" That is, where the F test focuses on the dependents, the multivariate tests focus on the independents and their ...