Small-Sample Sandwich Corrections for GEE

Why Adjust the Empirical Sandwich?

The empirical sandwich covariance estimates variability by adding the outer products of cluster-level estimating contributions. With few independent clusters, this empirical “meat” can be too small because the fitted coefficients have already used information from those same clusters. A high-leverage cluster, in particular, tends to have a reduced fitted residual. The Mancl–DeRouen (MD) and Kauermann–Carroll (KC) corrections address this effect by adjusting residuals according to cluster leverage.

The corrections differ in how strongly they inflate residual components in high-leverage directions. Neither is a universal remedy: both are finite-sample adjustments to a sandwich estimate, not guarantees of unbiased covariance estimation or accurate coverage.

Setup and Definitions

Let \(Y_i\) be the response vector for cluster \(i\), for \(i=1,\ldots,K\), and let \(\hat\beta\) be the fitted GEE coefficient vector of dimension \(p\). At the fitted values, write \(D_i\) for the derivative of the marginal mean vector with respect to \(\beta\), and \(V_i\) for the working covariance matrix. Assume each \(V_i\) is symmetric positive definite and the estimating-equation bread is nonsingular. Define

Here \(r_i\) is the fitted residual vector and \(s_i\) is the cluster estimating contribution. The ordinary empirical meat and sandwich covariance are

To define leverage adjustments without ambiguity about matrix square roots, use whitened cluster coordinates. Let \(V_i^{-1/2}\) be the symmetric positive-definite inverse square root of \(V_i\), and set

Then \(A=\sum_i Z_i^T Z_i\), and \(s_i=Z_i^T e_i\). The full projection matrix for the stacked whitened design has cluster-diagonal block \(P_{ii}=Z_iA^{-1}Z_i^T\). This symmetric matrix is the leverage matrix used below.

These inverses require every eigenvalue of \(P_{ii}\) to be strictly less than one. The formulas are plug-in corrections evaluated at the fitted model quantities; they do not alter the fitted GEE coefficients.

What the Leverage Eigenvalues Show

The leverage matrices are contractions. To see this, stack all \(Z_i\) into a matrix \(Z\). Since \(A=Z^TZ\) is nonsingular, \(P=ZA^{-1}Z^T\) is symmetric and idempotent, hence an orthogonal projection. The matrix \(P_{ii}\) is its principal block for cluster \(i\). For any vector \(x\) of the appropriate dimension, embed \(x\) in the stacked space as \(u\), with zeros in the other cluster positions. Then

Thus the eigenvalues \(h\) of \(P_{ii}\) lie in \([0,1]\); the inverse adjustments additionally require \(h<1\). Because the adjustments are functions of the same symmetric matrix, they act separately along its eigenvectors.

Indeed, putting \(t=1-h\), we have \(0<t\leq1\), so \(1/t\geq1/\sqrt{t}\). By the spectral theorem, this also means \((I-P_{ii})^{-1}-(I-P_{ii})^{-1/2}\) is positive semidefinite. Consequently, for a fixed residual vector, the MD adjusted residual has at least as large a Euclidean norm as the KC adjusted residual. This comparison concerns the residual multipliers themselves. It does not establish an ordering of the corrected meat matrices.

Why the Meat Matrices Need Not Be Ordered

A corrected meat contribution is an outer product of an adjusted score vector. Even if one adjustment matrix is larger in the positive-semidefinite ordering, that fact alone does not order the outer products of the resulting vectors. In particular, if two adjusted score vectors point in different directions, their outer products can differ by an indefinite matrix.

Near-Unit Leverage and Practical Use

Both corrections become increasingly aggressive as leverage approaches one, but at different rates. The MD multiplier grows as \((1-h)^{-1}\), while the KC multiplier grows as \((1-h)^{-1/2}\). Since the meat uses an outer product, the corresponding multiplier on a residual component’s squared contribution is \((1-h)^{-2}\) for MD and \((1-h)^{-1}\) for KC.

In practice, MD is the more aggressive of the two adjustments in each leverage eigen-direction, while KC applies the square-root adjustment and is less aggressive near unit leverage. That comparison is useful when explaining why estimates differ, but it is not a universal ranking of their performance. Small-cluster behavior depends on the design, cluster-size pattern, working covariance, and mean-model adequacy. A correction cannot make a misspecified mean model correct, and a nearly singular \(I-P_{ii}\) can make either estimate unstable.

For a small-cluster analysis, report which correction was used and examine whether the resulting standard errors are sensitive to the choice. Also inspect leverage and the influence of individual clusters; a dramatic increase can be a warning about limited information rather than evidence that one formula has solved the small-sample problem. These covariance adjustments alone do not specify the reference distribution or degrees of freedom for a test.

Common Mistakes

• Claiming a matrix ordering from multiplier ordering. The MD residual multiplier is larger in each leverage eigen-direction, but this does not imply that its adjusted score outer product exceeds the KC outer product in every direction. The difference can be indefinite.
• Comparing only the residual multipliers. The meat uses outer products, so a score multiplier is squared in a one-dimensional contribution. In several dimensions, the direction of the adjusted score also matters.
• Treating either correction as a guarantee. These are finite-sample covariance adjustments, not guarantees of unbiasedness, correct coverage, or stability.
• Ignoring leverage near one. The inverse adjustments become large as an eigenvalue approaches one. Check whether the fit is sufficiently informative for the correction to be numerically and substantively credible.