Greenhouse-Geisser Epsilon

Why Summarize Contrast Covariance?

With repeated measurements, the covariance among occasions affects inference about within-subject effects. Sphericity is the condition that the covariance is spherical in the within-subject contrast space. When the covariance is not spherical, a single scalar can summarize how far its contrast-coordinate eigenvalues are from being equal: the Greenhouse–Geisser epsilon. It is a property of the contrast covariance, not of the observed sample alone.

The key is to calculate epsilon in an orthonormal basis for the contrast space. This makes the trace formula invariant to a change of orthonormal basis and gives the familiar range from the reciprocal of the contrast-space dimension to one.

Definition in Orthonormal Contrast Coordinates

Suppose there are \(m\) repeated occasions, and let \(Y\) be the \(m\)-dimensional random vector of measurements, with covariance matrix \(\Sigma=\operatorname{Cov}(Y)\). Set \(d=m-1\). Let \(C\) be a \(d\times m\) matrix whose rows form an orthonormal basis for the contrasts among occasions. Thus \(CC'=I_d\) and \(C\mathbf{1}=0\), where \(\mathbf{1}\) is the \(m\)-vector of ones. The contrast response is \(Z=CY\), and its covariance matrix is

Assume \(S\) is nonzero and positive semidefinite. In the usual nondegenerate case, \(\Sigma\) is positive definite, so \(S\) is positive definite as well.

The formula compares the squared total contrast variance, \(\{\operatorname{tr}(S)\}^2\), with the sum of squared eigenvalues, \( \operatorname{tr}(S^2)\). Multiplying every contrast by the same nonzero constant multiplies the numerator and denominator by the same fourth-power factor, so epsilon is unchanged. A change to a different orthonormal contrast basis also leaves it unchanged.

Trace Formula and Basis Invariance

Define the centering projection \(P=I_m-\mathbf{1}\mathbf{1}'/m\), which projects onto the subspace of contrasts. Because the rows of \(C\) are an orthonormal basis of that subspace, \(C'C=P\). The trace formula can therefore also be written directly in terms of the occasion covariance matrix.

To verify the first identity, use \(S=C\Sigma C'\) and the cyclic property of the trace:

For the second identity, insert \(C'C=P\) between the two covariance factors, then cycle the trace:

If \(\widetilde C=QC\) for an orthogonal \(d\times d\) matrix \(Q\), then \(\widetilde C\) is another orthonormal contrast basis and its covariance is \(\widetilde S=QSQ'\). The trace is unchanged under this orthogonal conjugation: \(\operatorname{tr}(\widetilde S)=\operatorname{tr}(S)\), and \(\operatorname{tr}(\widetilde S^2)=\operatorname{tr}(QS^2Q')=\operatorname{tr}(S^2)\). Thus the definition does not depend on which orthonormal contrast basis is used.

Bounds and Equality Conditions

The bounds follow from the eigenvalues of the contrast covariance. Since \(S\) is symmetric and positive semidefinite, it has real eigenvalues \(\lambda_1,\ldots,\lambda_d\), all nonnegative. Its trace is their sum, and the trace of its square is the sum of their squares.

Proof. Write \(t=\sum_{i=1}^d\lambda_i\). Since \(S\) is nonzero and positive semidefinite, \(t>0\) and \(\sum_i\lambda_i^2>0\). By the Cauchy–Schwarz inequality,

Dividing by \(d\sum_i\lambda_i^2\) gives \(\epsilon_{\mathrm{GG}}\leq 1\). Equality in Cauchy–Schwarz holds exactly when the eigenvalues are all equal. For the other bound, nonnegativity gives

Dividing by \(d\sum_i\lambda_i^2\) gives \(\epsilon_{\mathrm{GG}}\geq 1/d\). Equality requires every product \(\lambda_i\lambda_j\), for \(i<j\), to be zero, so at most one eigenvalue can be nonzero. If \(S\) is positive definite and \(d>1\), all eigenvalues are positive, making at least one product positive; the lower inequality is then strict. \(\square\)

The upper equality condition is precisely equal variance in every orthonormal contrast direction, which is sphericity in contrast coordinates. The lower equality is a boundary case: all contrast variability is concentrated in at most one direction. For a positive-definite contrast covariance with more than one contrast, epsilon approaches but does not attain that lower bound.

Common Pitfall: Using Nonorthonormal Coordinates

The trace formula is not invariant under an arbitrary nonsingular change of contrast coordinates. For example, begin with \(S=I_2\), for which epsilon is \(1\). If the coordinates are changed using \(A=\operatorname{diag}(2,1)\), the transformed covariance is \(ASA'=\operatorname{diag}(4,1)\). Applying the formula to this matrix as though it came from an orthonormal basis gives

not \(1\). The error is treating the rescaled coordinates as orthonormal contrasts. To obtain the Greenhouse–Geisser epsilon, use an orthonormal contrast basis, or use the equivalent projection formula involving \(P\) and \(\Sigma\). This distinction matters even though the transformed coordinates span the same contrast space.

What Epsilon Summarizes

Epsilon measures the spread of contrast-covariance eigenvalues relative to their total. When they are equal, the covariance is spherical and epsilon is one. As the eigenvalues become more uneven, the sum of their squares becomes larger relative to the square of their sum, and epsilon decreases. Thus \(d\epsilon_{\mathrm{GG}}\) can be viewed as an effective number of equally variable contrast directions: it equals \(d\) under sphericity and is smaller when variability is concentrated in fewer directions.

The scalar is useful because it compresses a covariance structure into a single interpretable quantity while preserving the dependence on the contrast space. It does not identify which contrasts have high or low variance; distinct covariance matrices can have the same eigenvalue summary and hence the same epsilon. The full covariance matrix remains important when the direction of the departures from sphericity matters.