Theory and Methods of Statistics

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 = n₁ + ⋯ + n_k is the total number ...