= Multivariate analysis of variance
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= MANOVA
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Multivariate analysis of variance compares mean vectors of several groups while accounting for correlations between their response variables. In the one-way normal model, independent observations $Y_{ri}\sim N_p(\mu_r,\Sigma)$ share a positive-definite <covariance matrix>. Put $E=\sum_{r,i}(Y_{ri}-\bar Y_r)(Y_{ri}-\bar Y_r)^T$ and $H=\sum_rn_r(\bar Y_r-\bar Y)(\bar Y_r-\bar Y)^T$. Under equal means, these matrices have independent <Wishart distributions> with degrees of freedom $N-g$ and $g-1$, respectively. This follows by separating the Gaussian data into orthogonal within-group and between-group projections. The <Wilks lambda statistic> tests equality of the means using both matrices, rather than independently testing each coordinate.
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