Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 46 / 4 / i

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 $Y_{ri}\sim N_p({\mu }_r,\Sigma )$, $r=1,\dots ,g$, $i=1,\dots ,n_r$, with a common positive-definite covariance. Put $N={\sum }_r{n}_r$, and let ${\bar{Y}}_r$ and $\bar{Y}$ be group and overall sample means. The within-group and between-group scatter matrices areExpanding 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 statisticwhere ${\theta }_j$ 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 $N-g\ge p$. 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 iswith $N>p+1$. This formula demonstrates explicitly how the covariance weights a group difference.

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. MANOVA uses shared covariance to test a multivariate mean difference, rather than testing unrelated coordinates in isolation.