Noncentral Roots Under MANOVA Alternatives

Fixed Alternatives and Random Roots

Under a MANOVA null hypothesis, the hypothesis sum-of-squares matrix is central Wishart in the standard Gaussian linear model. Under a fixed alternative, the mean displacement contributes a systematic component, but sampling variation remains. The resulting hypothesis matrix is noncentral Wishart. MANOVA roots are then formed by comparing this random matrix with the error sum-of-squares matrix.

The noncentrality matrix records the size and direction of the fixed mean displacement after accounting for the design and response covariance. Its eigenvalues describe how that displacement is distributed across response directions. They are not themselves the MANOVA roots: roots also depend on the random error matrix and the hypothesis-matrix sampling variation.

The Noncentral Wishart Representation

Consider \(q\) hypothesis degrees of freedom and \(p\) response variables. After whitening the response covariance, represent the hypothesis matrix as \(H=(Z+M)^{\mathsf T}(Z+M)\), where \(Z\) is a \(q\)-by-\(p\) matrix of independent standard normal entries and \(M\) is a fixed \(q\)-by-\(p\) mean-shift matrix. The error matrix \(E\) is independent of \(H\), with \(E\sim W_p(\nu,I_p)\). Here \(\nu\) is the error degrees of freedom and \(I_p\) is the \(p\)-by-\(p\) identity matrix. Assume \(\nu\ge p\), so \(E\) is positive definite with probability one.

In an unwhitened linear model, let \(\Sigma\) be the response covariance, let \(D\) be the fixed mean of the tested contrasts, and let \(A\) be the positive-definite design weighting matrix for those contrasts. The corresponding whitened mean shift is \(M=A^{1/2}D\Sigma^{-1/2}\), and therefore \(\Omega=\Sigma^{-1/2}D^{\mathsf T}AD\Sigma^{-1/2}\). This form shows that the noncentrality is measured relative to both the design and the response covariance. Its rank is no larger than the rank of the tested mean displacement.

The eigenvalues of \(\Omega\) are unchanged by an orthogonal rotation of response coordinates. The trace \(\operatorname{tr}(\Omega)\) is the total squared whitened mean displacement. A few large eigenvalues describe a concentrated alternative; several smaller positive eigenvalues describe a more diffuse alternative. These features shape the hypothesis matrix, but should not be mistaken for a deterministic list of MANOVA roots.

Properties of the Noncentrality Matrix

The noncentrality matrix determines the noncentral Wishart law, not just the overall displacement magnitude. The following moment-generating function makes this dependence explicit. For symmetric \(T\) such that \(I_p-2T\) is positive definite, the matrix moment-generating function is defined here as \(E[\exp\{\operatorname{tr}(TH)\}]\).

Proof. Write the rows of \(Z+M\) as \(x_i^{\mathsf T}\), with corresponding fixed mean row vectors \(m_i^{\mathsf T}\), for \(i=1,\ldots,q\). Then \(\operatorname{tr}(TH)=\sum_{i=1}^q x_i^{\mathsf T}Tx_i\), and the rows are independent. For one row, \(x=m+z\) with \(z\sim N_p(0,I_p)\). Its Gaussian density gives

For the second equality, the quadratic term in \(z\) is \(-\tfrac12 z^{\mathsf T}(I_p-2T)z\). Completing the square is valid because \(I_p-2T\) is positive definite; integrating the resulting centered Gaussian contributes the determinant factor shown. Multiplying the \(q\) row contributions gives determinant power \(-q/2\). The exponent sums to \(\sum_i m_i^{\mathsf T}T(I_p-2T)^{-1}m_i =\operatorname{tr}\{\Omega T(I_p-2T)^{-1}\}\), since \(\Omega=\sum_i m_i m_i^{\mathsf T}=M^{\mathsf T}M\). This proves the formula. \(\square\)

The same model gives an immediate mean calculation. Expanding \(H\) yields \(H=Z^{\mathsf T}Z+Z^{\mathsf T}M+M^{\mathsf T}Z+M^{\mathsf T}M\). Each cross term has expectation zero, while \(E[Z^{\mathsf T}Z]=qI_p\). Hence

This identifies the average increase in total hypothesis sum of squares under the alternative. It does not say that an observed root equals an eigenvalue of \(E[H]\), or that a root distribution is determined by the trace alone.

How Alternative Structure Enters the Root Spectrum

For each realization, the roots are obtained from the pair \((H,E)\), with \(H\) noncentral Wishart and \(E\) independent central Wishart. The noncentrality eigenvalues encode the strengths of fixed alternatives along response directions. Since the root calculation uses the full matrices and the random error matrix, one should not replace the random roots by the noncentrality eigenvalues, nor infer their exact distribution from \(\operatorname{tr}(\Omega)\) alone.

A rigorous way to see that alternative structure can change a root spectrum is to examine a large-error-degrees-of-freedom limit. The next result relates the scaled generalized roots to the eigenvalues of \(H\). It also allows a root-spectrum comparison without assuming that different hypothesis-matrix laws automatically imply different root laws at a fixed \(\nu\).

Proof. Write \(E_\nu=\sum_{j=1}^{\nu} y_jy_j^{\mathsf T}\), where the \(y_j\) are independent \(N_p(0,I_p)\) vectors. Entrywise, the weak law of large numbers gives \(E_\nu/\nu\xrightarrow{p}I_p\). The generalized roots of \((H,E_\nu)\) are the ordinary eigenvalues of the symmetric matrix \(E_\nu^{-1/2}HE_\nu^{-1/2}\). Thus \(\nu\theta_{i,\nu}\) are the ordered eigenvalues of \((E_\nu/\nu)^{-1/2}H(E_\nu/\nu)^{-1/2}\). The map \(A\mapsto A^{-1/2}\) is continuous on positive-definite matrices, and the ordered eigenvalues of a symmetric matrix are continuous functions of its entries. Since \(H\) has a fixed distribution and is independent of \(E_\nu\), Slutsky's theorem and continuity imply that this matrix converges in distribution to \(H\), and its ordered eigenvalues converge in distribution to those of \(H\). This proves the claim. \(\square\)

Interpretation and a Common Pitfall

Under a fixed alternative, \(\Omega\) determines the noncentral Wishart component of the MANOVA construction, while \(q\), \(\nu\), and the random matrices determine how that component appears in the roots. Concentration in a few directions can produce a different root-spectrum behavior from displacement spread across directions, even when total noncentrality is held fixed. The exact effect on a chosen criterion requires examining its roots and calibration; a trace comparison alone is not a complete power analysis.

• Pitfall: “The noncentrality matrices differ, so the generalized-root distributions must differ.” This conclusion does not follow from differing distributions of \(H\): taking roots is a transformation that can discard information.
• Correction: State what the calculation proves. To make a claim about roots, derive their law or compare a quantity determined by their distribution. The limiting determinant comparison above does this for the scaled root spectra in the large-\(\nu\) regime.
• Pitfall: Treating the eigenvalues of \(\Omega\) as observed MANOVA roots. They describe fixed whitened mean displacement; the roots are random generalized eigenvalues of \(H\) relative to \(E\).