Local Alternatives and Asymptotic MANOVA Power

Local Alternatives and the Scale of MANOVA Roots

A useful way to compare MANOVA criteria is to study alternatives that become weaker as the sample size grows, but remain statistically visible. In the classical fixed-dimension regime, this means that the noncentrality matrix stays fixed while the residual degrees of freedom increase. The generalized roots then shrink, so comparing unscaled criterion values can conceal their common leading behavior.

Let \(p\) be the response dimension, \(m\) the numerator degrees of freedom, and \(q\) the residual degrees of freedom. Assume the hypothesis and error sum-of-squares-and-cross-products matrices are independent, with \(H\sim W_p(m,I_p,\Omega)\) and \(E\sim W_p(q,I_p)\). Here \(I_p\) is the \(p\times p\) identity matrix and \(\Omega\) is a fixed positive-semidefinite noncentrality matrix. As \(q\) increases, \(p\), \(m\), and \(\Omega\) are fixed. In a sequence of models, fixed \(\Omega\) represents effects shrinking at the rate that keeps noncentrality bounded.

For a Gaussian representation of \(H\), take a \(p\times m\) matrix \(Z\) with independent unit-variance normal entries and mean matrix \(M\), so \(H=ZZ'\) and \(MM'=\Omega\). This requires \(\operatorname{rank}(\Omega)\leq m\): a matrix \(M\) with \(m\) columns cannot have \(MM'\) of rank greater than \(m\). The condition holds for the usual MANOVA construction, in which the noncentrality is generated by a mean matrix with \(m\) columns. We impose it whenever using this representation.

Limiting Criteria and Power

The residual matrix grows with \(q\), while the hypothesis matrix remains on a fixed scale. Since \(E/q\) converges in probability to \(I_p\), the rescaled generalized roots converge to the eigenvalues of \(H\). This yields a common first-order limit for the three trace criteria, but not for Roy’s criterion.

Proof. For fixed \(p\), the entries of \(E\) are sums of \(q\) independent contributions with finite variances, so the law of large numbers gives \(E/q\xrightarrow{p}I_p\). The matrix \(E^{-1}H\) is similar to the symmetric matrix \(E^{-1/2}HE^{-1/2}\). Thus its roots are the eigenvalues of \((E/q)^{-1/2}H(E/q)^{-1/2}\). By continuity of the inverse square root at \(I_p\), this symmetric matrix differs from \(H\) by a quantity converging to zero in probability. The eigenvalues of symmetric matrices are continuous functions of their entries; consequently, the ordered rescaled roots \(q\lambda_i\) converge jointly in distribution to the ordered eigenvalues \(\mu_i\) of \(H\).

Write \(x_i=q\lambda_i\). Then \(qU=\sum_i x_i\), \(qV=\sum_i x_i/(1+x_i/q)\), and \(q(-\log\Lambda)=\sum_i q\log(1+x_i/q)\). For every bounded \(x\geq0\), \(x/(1+x/q)\to x\) and \(q\log(1+x/q)\to x\). The limiting random vector \((\mu_1,\ldots,\mu_s)\) is finite almost surely, so applying these continuous limits to the jointly converging roots gives the first three claims. Also \(qR=\max_i x_i\), which converges to \(\max_i\mu_i\). These arguments establish the joint limits. \(\square\)

The trace limit has a simple distribution. With \(H=ZZ'\), \(\operatorname{tr}(H)\) is the sum of the squares of all \(pm\) entries of \(Z\). A sum of squares of independent unit-variance normal variables with means whose squared sum is \(\operatorname{tr}(\Omega)\) has a noncentral chi-square distribution. Therefore \(\operatorname{tr}(H)\sim\chi^2_{pm}(\operatorname{tr}\Omega)\). For a level-\(\alpha\) upper-tail test based on any of the three scaled trace criteria, the limiting power is \(P\{\chi^2_{pm}(\operatorname{tr}\Omega)>c_\alpha\}\), where \(c_\alpha\) is the \(1-\alpha\) quantile of the central \(\chi^2_{pm}\) distribution.

To justify this power statement, under the null the scaled statistics converge to the central chi-square variable, whose distribution is continuous. Their level-\(\alpha\) upper-tail critical values therefore converge to \(c_\alpha\). Under the alternative, the same statistics converge to the stated noncentral chi-square variable, which also has a continuous distribution. The probability of exceeding the converging critical value consequently converges to the displayed tail probability. Roy’s limiting power instead uses a central-Wishart largest-root critical value and the alternative probability \(P\{\mu_1>r_\alpha\}\). It does not reduce to the trace chi-square calculation.

What the Noncentrality Spectrum Changes

The trace criteria’s limiting power depends on \(\Omega\) only through its trace. In particular, the eigenvalues of \(\Omega\)—the noncentrality spectrum—can be redistributed while keeping their sum fixed without changing this limiting trace power. Roy’s limit is different: it is the largest eigenvalue of a noncentral Wishart matrix, whose distribution reflects the directional structure of the alternative. It does not have a general reduction to a noncentral chi-square distribution indexed only by \(\operatorname{tr}(\Omega)\). Thus Roy retains spectral information that the limiting trace power discards.

The criteria also have different sensitivities as functions of a given root vector. If one root changes by a small amount while the others are fixed, the derivatives with respect to \(\lambda_i\) are \(1\) for \(U\), \(1/(1+\lambda_i)^2\) for \(V\), and \(1/(1+\lambda_i)\) for \(-\log\Lambda\). Roy responds to the largest root; when that root is unique, its derivative with respect to it is 1 and with respect to the other roots is 0. Under local alternatives the roots are of order \(1/q\), so both trace derivatives approach 1. This is the differential version of their shared first-order trace limit. At non-small roots, their weightings differ.

Rank Conditions and Common Pitfalls

The Gaussian representation places a concrete restriction on examples and derivations. If \(H=ZZ'\) and \(Z\) has \(m\) columns, then \(\operatorname{rank}(H)\leq m\), and likewise \(\operatorname{rank}(MM')\leq m\). A proposed noncentrality matrix with larger rank cannot arise from that representation. This matters especially when describing the distribution of the limiting Roy root, since that description is based on the noncentral Wishart matrix \(H\).

• Comparing unscaled statistics. Under these local alternatives, the roots shrink with \(q\). The nondegenerate comparisons use scaled criteria such as \(qU\), not \(U\) alone.
• Claiming all four criteria have the same limit. The three trace criteria share the trace limit; Roy converges to a largest eigenvalue.
• Equating equal trace with equal spectral sensitivity. Equal \(\operatorname{tr}(\Omega)\) gives equal limiting trace power, but does not establish equal Roy power.
• Ignoring the rank condition. A \(p\times m\) mean matrix cannot generate a noncentrality matrix of rank greater than \(m\).