The Central MANOVA Root Density

Why the Joint Root Density Matters

Under a central MANOVA null hypothesis, the hypothesis and error sums-of-squares-and-cross-products matrices fluctuate even when there is no population effect. Their generalized eigenvalues therefore have a nontrivial joint distribution. That distribution describes the roots together, rather than treating them as unrelated statistics, and is the starting point for exact null calculations.

The degrees of freedom determine both the shape of the density and how many roots are positive. In particular, a hypothesis matrix with fewer degrees of freedom than response dimensions is singular: it produces some roots that are exactly zero, as well as a smaller set of random positive roots.

Central Wishart Setup and Root Support

For the usual integer degrees of freedom, assume \(m\geq 1\) and \(\nu\geq p\). The latter condition makes \(E\) positive definite almost surely, so the generalized eigenvalue problem is well-defined. The hypothesis matrix has rank \(\min(p,m)\) almost surely. We study only its \(q\) positive roots below. Unless roots are explicitly ordered, their density is for an unordered vector: it is symmetric in the \(q\) coordinates and integrates to one over the full positive orthant.

It is useful to map a positive root to \(x=\lambda/(1+\lambda)\). These transformed roots lie between zero and one. They are the eigenvalues associated with \(H\) relative to \(H+E\); this bounded support makes the joint density especially simple. The transformation does not change the number of positive roots.

The Joint Density

The absolute-difference product is essential: roots repel in the joint density, and the density is not a product of \(q\) independent one-dimensional densities. The exponent at zero depends on the difference between response dimension and hypothesis degrees of freedom. The common power of \(1+\lambda_i\) depends on the combined hypothesis and error degrees of freedom.

Derivation from Wishart Matrices

First consider the transformed roots \(x_i=\lambda_i/(1+\lambda_i)\). Write \(a=|p-m|\) and \(b=\nu-p\). Their joint density on \((0,1)^q\) is

Here is the Wishart argument for this Jacobi-form density. Let \(X\) be an \(m\times p\) matrix and \(Y\) a \(\nu\times p\) matrix, all of whose entries are independent standard normal variables, with the two matrices independent. Then \(H=X^\mathsf{T}X\) and \(E=Y^\mathsf{T}Y\) have the required independent central Wishart distributions. If \(Z\) is the matrix formed by stacking \(X\) above \(Y\), then

has orthonormal columns and is uniformly distributed over such \(p\)-frames in \(\mathbb{R}^{m+\nu}\). Let \(P\) be the projection onto the first \(m\) coordinates. The eigenvalues of \(U^\mathsf{T}PU\) are precisely the roots of \(H\) relative to \(H+E\), including any zero roots.

The block-coordinate change for a uniform orthonormal frame gives the following density for its nonzero eigenvalues. When \(m\geq p\), the \(p\times p\) matrix \(U^\mathsf{T}PU\) has density proportional to \(|C|^{(m-p-1)/2}|I_p-C|^{(\nu-p-1)/2}\) on \(0<C<I_p\). When \(m<p\), use the \(m\times m\) matrix \(U_1U_1^\mathsf{T}\), where \(U_1\) comprises the first \(m\) rows of \(U\); it has the same nonzero eigenvalues and density proportional to \(|C|^{(p-m-1)/2}|I_m-C|^{(\nu-p-1)/2}\). These powers are the determinant powers in the frame-coordinate Jacobian. In both cases, diagonalizing the symmetric matrix contributes the eigenvalue Jacobian \(\prod_{i<j}|x_i-x_j|\). Thus the powers in the displayed density are \((|p-m|-1)/2\) at zero and \((\nu-p-1)/2\) at one.

The integral that normalizes this density is the Selberg integral on \((0,1)^q\). In its real case, with the Vandermonde power equal to one, it is exactly \(S_q(\alpha,\beta,1/2)\) given in the theorem. This establishes both the density and its normalization for the transformed roots.

It remains to change variables back to \(\lambda_i\). Since \(x_i=\lambda_i/(1+\lambda_i)\),

Also, \(x_i^{(a-1)/2}=\lambda_i^{(a-1)/2}(1+\lambda_i)^{-(a-1)/2}\) and \((1-x_i)^{(b-1)/2}=(1+\lambda_i)^{-(b-1)/2}\). For each \(\lambda_i\), the Vandermonde contributes a further power \(-(q-1)\) of \(1+\lambda_i\). The total exponent in the denominator is therefore

because \(a+2q=p+m\) and \(b=\nu-p\). The change of variables proves the stated root density, including its support and constant. \(\square\)

Worked Examples

Common Pitfall and Use of the Density

• Confusing the two supports. The transformed roots \(x_i\) lie in \((0,1)\), but the generalized roots \(\lambda_i=x_i/(1-x_i)\) range over all positive values. Applying the bounded support to \(\lambda_i\) is incorrect.
• Counting zero roots as random positive roots. When \(m<p\), the density has \(q=m\) coordinates. The other \(p-m\) roots are identically zero, not additional draws from this density.
• Assuming roots are independent. The factor \(\prod_{i<j}|\lambda_i-\lambda_j|\) couples the roots. Separate marginal densities cannot be multiplied to obtain their joint law.
• Dropping the normalization convention. The constant above is for unordered roots over the entire positive orthant. If roots are instead restricted to an ordering, the density and its normalizing constant must be adjusted for that restricted domain.

The density matters because exact null calculations depend on the joint root law, including the dependence induced by the Vandermonde factor. Its degrees-of-freedom conditions also clarify when an analysis has a full set of positive roots and when rank forces zero roots. The Selberg form provides a direct normalization check before using the roots in a MANOVA criterion.