Pairwise Difference Characterization of Sphericity

Why Pairwise Differences Characterize Sphericity

In a repeated-measures setting, sphericity concerns variation in contrasts among occasions, rather than variation in the overall level of a subject's measurements. A direct way to describe that contrast variation is to compare every pair of occasions. This leads to a useful characterization: the contrast covariance is spherical precisely when every pairwise difference has the same variance. The strict positivity condition matters, however. Equal difference variances can have the common value zero, which corresponds to a singular contrast covariance rather than ordinary positive sphericity.

Let \(Y=(Y_1,\ldots,Y_q)'\) be a random vector of measurements on \(q\geq 2\) occasions, with finite second moments and covariance matrix \(\Sigma=\operatorname{Cov}(Y)\). Write \(\mathbf{1}\) for the \(q\)-vector of ones. A contrast is a linear combination whose coefficients sum to zero; the contrast space is the set of vectors \(a\) satisfying \(\mathbf{1}'a=0\). Choose a \(q\times(q-1)\) matrix \(C\) whose columns form an orthonormal basis of that space. Thus \(C'C=I_{q-1}\) and \(C'\mathbf{1}=0\).

The definition does not depend on which orthonormal contrast basis is chosen. If \(C_1\) and \(C_2\) are two such bases, then \(C_2=C_1Q\) for an orthogonal \((q-1)\times(q-1)\) matrix \(Q\), so \(C_2'\Sigma C_2=Q'(C_1'\Sigma C_1)Q\). In particular, a scalar multiple of the identity in one basis is the same scalar multiple in every orthonormal contrast basis.

The Pairwise-Difference Characterization

The key geometric object is the projection onto the contrast space, \(P=I_q-\mathbf{1}\mathbf{1}'/q\). It satisfies \(P'=P\), \(P^2=P\), \(P\mathbf{1}=0\), and \(C C'=P\). For any pair, \(d_{ij}=e_i-e_j\) is already a contrast, so \(Pd_{ij}=d_{ij}\). Consequently its variance depends only on the covariance restricted to the contrast space: \(d_{ij}'\Sigma d_{ij}=d_{ij}'P\Sigma P d_{ij}\).

Proof. First suppose \(C'\Sigma C=\lambda I_{q-1}\) for some \(\lambda\geq0\). Since \(d_{ij}\) lies in the contrast space, \(d_{ij}=C C'd_{ij}\), and \(C'd_{ij}\) has squared length \(d_{ij}'C C'd_{ij}=d_{ij}'P d_{ij}=d_{ij}'d_{ij}=2\). Therefore

This gives the same difference variance for every pair, with \(v=2\lambda\).

For the converse, suppose every pairwise difference variance is \(v\geq0\). Set \(A=P\Sigma P\), a symmetric matrix satisfying \(A\mathbf{1}=0\), and define \(B=A-(v/2)P\). Because \(d_{ij}'P d_{ij}=2\), the equal-variance assumption gives

Writing \(B_{ij}\) for the entries of \(B\), this means \(B_{ii}+B_{jj}-2B_{ij}=0\) whenever \(i\ne j\). Hence \(B_{ij}=(B_{ii}+B_{jj})/2\). This formula also holds when \(i=j\), so it can be summed over each row. Since \(B\mathbf{1}=0\), for each \(i\),

Thus \(B_{ii}=-\operatorname{tr}(B)/q\) for every \(i\). Summing these \(q\) equalities gives \(\operatorname{tr}(B)=-\operatorname{tr}(B)\), so \(\operatorname{tr}(B)=0\). All diagonal entries are therefore zero, and the entry formula then gives \(B_{ij}=0\) for every \(i,j\). It follows that \(A=(v/2)P\). Finally, \(C'P C=I_{q-1}\) and \(C'P=C'\), so

This proves the equivalence, including the boundary case \(v=0\). Since \(\Sigma\) is a covariance matrix, the contrast covariance is positive semidefinite, so its scalar scale cannot be negative. It is strictly positive exactly when \(v>0\). \(\square\)

Worked Covariance Checks

Why the Result Is Useful—and a Common Pitfall

The characterization gives an interpretation of contrast sphericity that does not depend on a chosen contrast basis: all occasion pairs must have the same variability in their differences. It also supplies a direct algebraic check. Given \(\Sigma\), calculate \(\Sigma_{ii}+\Sigma_{jj}-2\Sigma_{ij}\) for each distinct pair. Equality of these quantities is equivalent to a scalar contrast covariance, and their common value is twice its scalar scale.

A common mistake is to conclude that equal pairwise difference variances always imply positive sphericity. The reasoning may correctly derive \(C'\Sigma C=(v/2)I\), but then silently treat \(v/2\) as strictly positive. The derivation only gives \(v\geq0\), because variances cannot be negative. The correction is to distinguish the two statements: for the singular extension, equal differences with \(v\geq0\) are equivalent; for positive sphericity, the common value must satisfy \(v>0\). A zero value is not a contradiction—it means all contrast directions have zero variance.