## 12.6 One-Factor MANOVA

Suppose that we have k multivariate normal populations with possibly different mean vectors, but the same covariance matrix. Let {Y_{ij}: j = 1, …, n_{i}} be iid N_{p}(μ_{i}, Σ), i = 1, …, k. We may write the one-factor MANOVA model as

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

where {ε_{ij}} are iid N_{p}(0, Σ). We can also rewrite the above as a factor-effect model

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

where $\mathit{\mu}=\sum ({n}_{i}/n){\mathit{\mu}}_{i}$, α_{i} =μ_{i} −μ and n = n_{1} + ⋯ + n_{k} is the total number ...

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