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For independent multivariate normal observations with positive-definite covariance, this statistic is the maximum squared studentized mean contrast. With unbiased sample covariance matrix $S$ and $n>p$, its null distribution obeys $(n-p)T^2/(p(n-1))\sim F_{p,n-p}$. This follows from independence of the Gaussian sample mean and the residual Wishart distribution.

If $W\sim W_p(\nu ,R^{-1})$ has the Wishart distribution, then $W^{-1}$ has the inverse-Wishart distribution. For $\nu >p+1$, its expected value is $R/(\nu -p-1)$; below this threshold the matrix mean does not exist. It commonly models a covariance matrix, but couples the prior behavior of variances and correlations.

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 $H={\sum }_r{n}_r({\bar{Y}}_r-\bar{Y})({\bar{Y}}_r-\bar{Y}{)}^T$. 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.

Multivariate analysis of variance tests differences between group mean vectors when each experimental unit has several correlated responses. For independent $p$-dimensional observations in $g$ groups, use a common positive-definite covariance matrix $\Sigma$ and the multivariate normal model $Y_{ri}\sim N_p({\mu }_r,\Sigma )$. The null hypothesis is equality of all $g$ means.

Write $n={\sum }_r{n}_r$, group means ${\bar{y}}_r$, and the size-weighted grand mean $\bar{y}$. The within-group and between-group scatter matrices areExpanding deviations about the group means yields total scatter $T=E+H$, with zero cross terms. Under the null, orthogonal normal projections give independent Wishart distributions $E\sim W_p(\Sigma ,n-g)$ and $H\sim W_p(\Sigma ,g-1)$.

With nonsingular $E$, the Wilks lambda statistic iswhere the nonnegative ${\theta }_j$ are eigenvalues of $E^{-1/2}H{E}^{-1/2}$. Maximizing the common covariance under the two mean models gives the likelihood ratio ${\Lambda }^{n/2}$, so small $\Lambda$ indicates large group separation relative to within-group spread. For fixed $p,g$ and large group sizes, the Wilks theorem gives $-n\log \Lambda \Rightarrow {\chi }_{p(g-1)}^2$ under the null; a finite-sample reference distribution can also be used.

The lower-right sketch shows two elongated groups whose centroid difference lies nearly across, rather than along, their long axes. Their correlation makes this multivariate separation much clearer than either marginal mean shift alone. Nonsingular linear rescaling changes $E$ and $H$ by congruence, leaving the determinant ratio unchanged. MANOVA measures mean separation against the full within-group covariance geometry. Its interpretation requires independent observations and an adequate common-covariance model; separate coordinatewise tests discard that geometry and raise multiple-testing issues.

With the divisor-$n$ sample covariance matrix defined above, the maximum-likelihood estimators areTheir joint sampling distribution is specified byHere $W_p(\nu ,V)$ uses the Wishart distribution convention $\mathbb{E}W=\nu V$. Thus the covariance maximum-likelihood estimator has expected value $(n-1)V/n$; the unbiased estimator of the covariance matrix is $nS/(n-1)$. These distributions follow from orthogonal projections of the jointly multivariate normal observations onto the constant-observation direction and its orthogonal complement, though no proof is required here. The covariance maximum in the positive-definite parameter space requires $S$ nonsingular, which holds almost surely when $n>p$ and $V$ is positive definite. Otherwise the rank of a centered sample covariance matrix shows why the usual unrestricted finite maximum does not exist.

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.

Model B replaces independent tree parameters by multivariate normal random effects. It permits correlated asymptote, timing and slope variation and supplies partial pooling, a useful improvement if the five trees can be regarded as exchangeable.

The Wishart distribution prior is on the precision matrix $\Omega$, not the covariance matrix. In the WinBUGS convention, dimension three and degrees of freedom three give a proper but very dispersed prior. The induced inverse-Wishart distribution has no finite mean at those degrees of freedom. Thus this is not a neutral choice of covariance prior, and estimating six covariance components from only five trees is weakly informed. The broad mean priors and possible negative growth slopes also remain. Check prior-predictive curves and sensitivity, and consider separate, interpretable scale and correlation priors.

The resulting posterior means of the second and third parameters are quite similar across trees, motivating a simpler shared-shape model; their uncertainty means this similarity is evidence to explore, rather than a proof of exact equality.