## 12.6 One-Factor MANOVA

Suppose that we have k multivariate normal populations with possibly different mean vectors, but the same covariance matrix. Let {Yij: j = 1, …, ni} be iid Np(μiΣ), i = 1, …, k. We may write the one-factor MANOVA model as

$\begin{array}{l}\hfill {\mathbit{Y}}_{ij}={\mathbit{\mu }}_{i}+{\mathbit{\epsilon }}_{ij},j=1,\dots ,{n}_{i},i=1,\dots ,k,\end{array}$

where {εij} are iid Np(0Σ). We can also rewrite the above as a factor-effect model

$\begin{array}{l}\hfill {\mathbit{Y}}_{ij}=\mathbit{\mu }+{\mathbit{\alpha }}_{i}+{\mathbit{\epsilon }}_{ij},\end{array}$

where $\mathbit{\mu }=\sum \left({n}_{i}/n\right){\mathbit{\mu }}_{i}$, αi =μiμ and n = n1 + ⋯ + nk is the total number ...

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