= Solution
<Multivariate analysis of variance> extends <analysis of variance> to vector responses. For example, a treatment may affect several correlated physiological measurements jointly. Separate univariate tests do not use the <correlation> geometry and require adjustment if they are interpreted as one multiple-testing procedure. A joint test asks whether group mean vectors differ in any direction.
In a one-way model, take <independent> $Y_{ri}\sim N_p(\mu_r,\Sigma)$, $r=1,\ldots,g$, $i=1,\ldots,n_r$, with a common <positive-definite> <covariance>. Put $N=\sum_rn_r$, and let $\bar Y_r$ and $\bar Y$ be group and overall <sample means>. The within-group and between-group scatter <matrices> are
$$
E=\sum_{r,i}(Y_{ri}-\bar Y_r)(Y_{ri}-\bar Y_r)^T,\qquad H=\sum_rn_r(\bar Y_r-\bar Y)(\bar Y_r-\bar Y)^T.
$$
Expanding around each group mean makes the cross terms vanish, giving total scatter $E+H$. To test $H_0:\mu_1=\cdots=\mu_g$, use the <Wilks lambda statistic>
$$
\boxed{\Lambda=\frac{|E|}{|E+H|}=\prod_{j=1}^{p}(1+\theta_j)^{-1},}
$$
where $\theta_j$ solve the <generalized eigenvalue problem> $Hv=\theta Ev$ and $E$ is nonsingular. Under the alternative, maximize the normal <likelihood> at the group means and <covariance> $E/N$; under the null, use the overall mean and <covariance> $(E+H)/N$. Their <likelihood ratio> is $\Lambda^{N/2}$, so small $\Lambda$ rejects equal means. Under the null, orthogonal projections of the Gaussian data give <independent> $E\sim W_p(\Sigma,N-g)$ and $H\sim W_p(\Sigma,g-1)$, supplying its null calibration. Here $E$ is nonsingular almost surely when $N-g\geq p$. Merely having more total observations than responses is not enough if group fitting consumes too many <statistical degrees of freedom>.
The test is invariant under a nonsingular linear change of response coordinates: both <determinants> acquire the same factor $|A|^2$. The directions associated with large $\theta_j$ identify mean separation relative to residual variability. With one response it becomes the ordinary <ANOVA> <F-test>. With two groups, writing $S_p=E/(N-2)$ and $d=\bar Y_1-\bar Y_2$, the version using <Hotelling's T-squared statistic> is
$$
T^2=\frac{n_1n_2}{N}d^TS_p^{-1}d,\qquad\frac{N-p-1}{p(N-2)}T^2\sim F_{p,N-p-1}\quad(H_0),
$$
with $N>p+1$. This formula demonstrates explicitly how the <covariance> weights a group difference.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-46-manova-geometry.png]
{title=Correlated response clouds can overlap marginally yet separate along a low-variance contrast}
The sketch shows hypothetical groups with strongly positively correlated responses and mean separation along their low-variance contrast. Marginal comparisons can look weak even when the joint comparison is informative. The ellipses represent individual-response <covariance>, not confidence regions for the means. <Independence> of observations, approximate multivariate normality and common <covariance> are substantive assumptions; outliers and unequal <covariance> can invalidate the usual calibration. Follow-up contrasts can identify the affected responses or groups, with appropriate multiplicity control. \b[<MANOVA> uses shared <covariance> to test a multivariate mean difference, rather than testing unrelated coordinates in isolation.]
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