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Since both your $X$ and $Y$ are rank deficient, you cannot directly compute the Hotteling $T$ because the $W$-matrix is singular. So, you have to replace your $X$ and $Y$ by their reduced rank approximations, which I'll denote $X^k$ and $Y^k$. where $$X^k=(X-\bar{X})V[(X-\bar{X})/\sqrt{n_x-1}]$$ where $\bar{X}=n_{x}^{-1}\sum_{i=1}^{n_x} X_i$ and ...

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Unless I am missing something, this can be seen from using Cauchy-Schwarz & Spectral Decomposition (as shown in pages 78-80 in Applied Multivariate Statistical Analysis, by Richard A. Johnson and Dean W. Wichern, 6th edition). First Cauchy-Schwarz: For two $p \times 1$ vectors $\mathbf{b}$ and $\mathbf{d}$,  (\mathbf{b}' \mathbf{d})^2 \leq ...

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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 ...

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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, $\| \{\lambda_1, \ldots, \lambda_p\} ... 1 My question was about the distributions, not the test, and I think I've figured out the answer: a t-distribution squared has an F(1,n) distribution, which is a Hotelling distribution (up to rescaling by a constant determined by the parameters). I believe one can say that an F(m,n) distribution is the same as a Wilks'$\Lambda(1,m,n)\$ distribution, which is ...

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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 ...

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