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Under a nonsingular affine transformation $Y=AX+b$, transform the null mean by the same map. Then the centered sample mean becomes $A(\bar{X}-{\mu }_0)$ and the sample covariance matrix becomes $AS{A}^T$. Since $(AS{A}^T{)}^{-1}=A^{-T}{S}^{-1}{A}^{-1}$, the quadratic form in Hotelling's T-squared statistic is unchanged. Its value and test decision are consequently invariant under changes of units, origin and nonsingular coordinates.

For a fixed rank-$m$ constraint matrix $B$, apply Hotelling's T-squared statistic to the transformed observations $B{X}_j$. Under $B\mu =0$, $(n-m)T_B^2/(m(n-1))\sim F_{m,n-m}$ for $n>m$. If the displayed contrasts are dependent, use a basis of their row space and replace $m$ by the rank.

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.

Assume an independent Gaussian sample, positive-definite population covariance, and $n>p$. Define the sample mean and unbiased sample covariance matrix byThe standard test is based on Hotelling's T-squared statisticReject the null hypothesis for large values. Under the null its exact distribution isThe factor uses the unbiased, divisor-$n-1$ covariance convention. Normal-sample theory makes $\bar{X}$ independent of the residual covariance, whose whitened version has a Wishart distribution; this yields the stated F-distribution. The condition $n>p$ gives invertibility of $S$ almost surely and positive denominator degrees of freedom.

This is also the likelihood-ratio test. To see the connection, put $W=(n-1)S$ and $\delta =\bar{X}-{\mu }_0$. The maximized covariance estimates under the unrestricted and null models are $W/n$ and $W/n+\delta {\delta }^T$, respectively. The determinant identity for a rank-one update givesThus the likelihood ratio is $(1+T^2/(n-1){)}^{-n/2}$, decreasing in $T^2$. The term best is understood here as this standard likelihood-ratio, affine-invariant procedure; a two-sided multidimensional alternative does not specify a single direction to optimize power against.

For the transformed sample $Y_i=A{X}_i+b$, with $A$ the prescribed nonsingular coordinate transformation, we haveConsequentlyThis proves the affine invariance of Hotelling's statistic. The statistic, its p-value and its test decision are unchanged by a nonsingular affine transformation, provided the null mean is transformed with the data. Changes of units and rotations are included.

Use the unbiased sample covariance matrixand assume $\Sigma$ is positive-definite and $n>p$, so $S$ is invertible almost surely. For a nonzero fixed direction $a$, the observations $Y_j=a^T{X}_j$ have mean $a^T\mu$ and variance $a^T\Sigma a$. Their squared Student t-test statistic for the projected null mean isPut $b=\bar{X}-{\mu }_0$. The Cauchy-Schwarz inequality giveswith equality when $a$ is proportional to $S^{-1}b$, if $b\ne 0$. For $b=0$ every projected statistic is zero. Maximizing the squared discrepancy over directions therefore gives Hotelling's T-squared statistic:This uses the squared generalized Rayleigh quotient. The preliminary formula on the printed cover has an unsquared numerator and is not valid as written: setting $a=t{C}^{-1}b$ makes that ratio $1/t$ for $b\ne 0$, so it is unbounded. The displayed Cauchy-Schwarz argument supplies the needed corrected identity independently.

Under the null, let $Z=\sqrt{n}(\bar{X}-{\mu }_0)$ and $C=(n-1)S$. Then $Z\sim N_p(0,\Sigma )$, $C\sim W_p(n-1,\Sigma )$, and $Z,C$ are independent. To see these sample properties, apply an orthogonal transformation to the $n$ observation indices with first row $n^{-1/2}(1,\dots ,1)$. Its first centered row is $Z$; its other $n-1$ rows are independent $N_p(0,\Sigma )$ residual contrasts, independent of the first. Their outer products sum to $C$, establishing the Wishart distribution and independence.

Since $T^2=(n-1)Z^T{C}^{-1}Z$, use the supplied Gaussian-Wishart quadratic-form law with $k=n-1$ to obtainThus a level-$\eta$ test rejects when this scaled statistic exceeds the $(1-\eta )$ quantile of the F-distribution. The Gaussian-Wishart law requires independence; it holds here because of the orthogonal sample decomposition. Its printed use of $X$ instead of the introduced $Z$ is a notational slip. A fixed-direction Student law alone would not calibrate the maximized statistic, because its maximizing direction depends on the sample.