Multivariate analysis of variance

Definition

Let $g \ge 2$ and $p \ge 1$ be integers. For $i = 1, \ldots, g$ and $j = 1, \ldots, n_i$, let $X_{ij}$ be independent random vectors in $\R^p$ with $X_{ij} \sim N_p(\mu_i, \Sigma)$, a multivariate normal distribution with unknown $\mu_i$ and unknown common $\Sigma$. Let $N = n_1 + \cdots + n_g$, let $\bar X_i$ be the mean of group $i$ and $\bar X$ the mean of all $N$ observations. The between-group and within-group matrices are $$B = \sum_{i=1}^g n_i (\bar X_i - \bar X)(\bar X_i - \bar X)^T, \qquad W = \sum_{i=1}^g \sum_{j=1}^{n_i} (X_{ij} - \bar X_i)(X_{ij} - \bar X_i)^T.$$ One-way multivariate analysis of variance is the test of $H_0 \colon \mu_1 = \cdots = \mu_g$ by a statistic that depends on the eigenvalues of $W^{-1} B$. The standard statistics are:

1. Wilks' lambda $\Lambda = \det W / \det(W + B)$, which rejects for small values.
2. Pillai's trace $\tr(B (B + W)^{-1})$.
3. The Hotelling–Lawley trace $\tr(W^{-1} B)$.
4. Roy's largest root, the largest eigenvalue of $W^{-1} B$.