Redundant Contrasts and Effective Family Rank

Why Redundant Contrasts Need Careful Interpretation

A simultaneous procedure may list many contrast targets even though those targets vary within a lower-dimensional subspace. This happens, for example, when one contrast is a linear combination of others. The dependence matters: it determines how many independent directions Scheffé’s procedure must protect, and it makes the correlation matrix used in max-t calculations singular. But dependence does not mean that every listed target can simply be deleted. A finite family of contrasts and the entire subspace they span are different objects.

Let \(X_1,\ldots,X_n\) be independent observations from \(N_p(\mu,\Sigma)\), with \(\Sigma\) positive definite. Write \(\bar X\) for their sample mean, \(S\) for their sample covariance matrix, and \(\nu=n-1\). Let \(C\) be an \(m\times p\) matrix whose rows are the contrast vectors \(c_j'\). The \(j\)th target is \(c_j'\mu\), and the listed target vector is \(C\mu\).

A zero row is not a genuine contrast target: it has zero variance and its target is identically zero. In what follows, the listed rows are nonzero unless a zero row is explicitly discussed. Since \(\Sigma\) is positive definite, each nonzero contrast has positive variance \(c_j'\Sigma c_j\).

Finding the Rank and Reading the Relations

The rank can be found by ordinary row reduction. It counts independent contrast directions, not necessarily the number of separately named targets. A row relation also gives a relation among the target values: if \(a'C=0\), then \(a'(C\mu)=0\). That is a statement about the targets for every possible mean vector, rather than a numerical coincidence in one sample.

Scheffé Protection Uses the Span Dimension

Scheffé protection concerns every contrast direction in the row space of \(C\), not merely the \(m\) rows printed in the matrix. The relevant dimension is therefore \(r\). The finite-sample calibration follows by applying the one-sample Hotelling \(T^2\) result to coordinates on that \(r\)-dimensional subspace.

Derivation. Choose a \(p\times r\) matrix \(B\) with full column rank whose columns form a basis for the column space of \(C'\). Every contrast vector in that space has the form \(c=Ba\) for some \(a\in\mathbb R^r\). The observations \(B'X_i\) are independent \(r\)-variate normal vectors with mean \(B'\mu\) and positive definite covariance \(B'\Sigma B\). Their sample mean and sample covariance are \(B'\bar X\) and \(B'SB\).

For any nonzero \(a\), the ratio in the theorem is \(\left|a'B'(\bar X-\mu)\right|/\sqrt{a'B'SBa/n}\). The supremum of its square over nonzero \(a\) is

The equality follows by writing \(u=B'(\bar X-\mu)\) and \(A=B'SB\): Cauchy–Schwarz in the inner product induced by \(A\) gives \((a'u)^2\le (a'Aa)(u'A^{-1}u)\), with equality when \(a\) is proportional to \(A^{-1}u\). The right-hand side is Hotelling’s \(T^2\) statistic for the \(r\)-dimensional observations \(B'X_i\). Under the stated normal model, its distribution is

Consequently, the probability that this quadratic form is at most \(r\nu F_{r,\nu-r+1;1-\alpha}/(\nu-r+1)\) is \(1-\alpha\). On that event, every direction has squared ratio at most \(rF_{r,\nu-r+1;1-\alpha}\), which is exactly the simultaneous inequality claimed. The argument requires \(r\le\nu\), so the reduced sample covariance is nonsingular with probability one. \(\square\)

Rank and Correlation in Max-t Calculations

For a finite family, a max-t procedure protects the listed contrasts rather than every direction in their span. To understand the role of rank, first consider the Gaussian contrast numerators \(c_j'(\bar X-\mu)\). Their covariance matrix is \(C\Sigma C'/n\). If each contrast is standardized by its population standard deviation, the resulting correlation matrix is

Because all rows are nonzero and \(\Sigma\) is positive definite, \(D\) is invertible and \(\operatorname{rank}(R)=\operatorname{rank}(C)=r\). In detail, for any \(x\in\mathbb R^m\),

which is zero exactly when \(C'x=0\). Thus \(C\Sigma C'\) and \(C'\) have the same null space in \(\mathbb R^m\), and multiplying on both sides by the invertible \(D^{-1}\) does not change rank. When \(m>r\), \(R\) is singular: its Gaussian vector is supported on an \(r\)-dimensional subspace of \(\mathbb R^m\).

This is the relevant algebra for max-t correlation calculations. In a max-t model with a common denominator, the vector can be represented as \(T=Z/\sqrt{U/\nu}\), where \(Z\sim N_m(0,R)\), \(U\sim\chi^2_\nu\), and \(Z\) and \(U\) are independent. Singular \(R\) causes no ambiguity: generate \(Z\) in its \(r\)-dimensional support, or factor \(R\) using its nonzero eigenvalues. The maximum is still taken over the listed coordinates. If a particular model uses contrast-specific estimated denominators, its exact joint distribution need not be this common-denominator multivariate \(t\) law; rank alone does not determine that distribution.

What Can Be Removed, and What Cannot

A specific finite-family simplification is valid when two rows are proportional. If \(c_k=q c_j\) for a nonzero scalar \(q\), their two-sided standardized statistics have identical absolute values whenever they use the same standard-error rule:

One of these rows may therefore be removed from a two-sided max without changing the maximum. This is a stronger condition than general linear dependence. For example, \(e_1+e_2\) lies in the span of \(e_1\) and \(e_2\), but it is not proportional to either, so its absolute statistic is generally a distinct term in the maximum.

The distinction also matters for calibration. Scheffé protection uses the dimension of a continuum of directions, and its multiplier uses \(r\). A finite max-t procedure uses the joint distribution of its particular listed statistics; its critical value depends on their dependence structure, not merely on the rank. A singular correlation matrix is a feature to handle in computation, not a reason to substitute \(r\) for the number of coordinates in a generic finite maximum.