Efficiency Under Working-Correlation Misspecification

Why a Correct Mean Does Not Guarantee Efficient Estimation

Generalized estimating equations (GEE) use a model for the marginal mean and a working covariance to combine observations within a cluster. The mean model can be correct even when the chosen working correlation is not. In that case, the estimating equation can still target the correct regression parameter, but its uncertainty depends on both the working covariance and the true covariance of the responses.

This distinction matters when choosing between working structures. A more flexible structure can represent more patterns of dependence, but flexibility alone does not ensure that its fitted weights are closer to the truth. If those weights are poor for the covariate pattern being estimated, the resulting estimator can be less efficient than one using a simpler working correlation.

The Sandwich Covariance Under Misspecification

Let \(Y_i\) be the response vector for independent cluster \(i\), and let its correctly specified marginal mean be \(\mu_i(\beta)\), with true parameter \(\beta_0\). Write \(D_i(\beta)=\partial\mu_i(\beta)/\partial\beta^\mathsf{T}\). The GEE uses a working covariance \(W_i\), which need not equal the true covariance \(V_i=\operatorname{Var}(Y_i)\). For now, take the working weights as fixed, or as converging to fixed limits.

Here the expectation denotes a common-cluster expectation; with non-identically distributed clusters, the corresponding limits of the averages of the cluster matrices are used. The result requires independent clusters, a fixed-dimensional parameter, a nonsingular limiting \(A\), finite second moments for the estimating functions, and regularity sufficient for a consistent root of the GEE, a first-order Taylor expansion, and a cluster-level central limit theorem. Cluster sizes are assumed bounded, or otherwise subject to conditions that ensure those limits and the central limit theorem. If working-correlation parameters are estimated, the same expression applies when their plug-in limits and regularity conditions give this first-order expansion.

The two matrices have different jobs. \(A\) describes how rapidly the expected estimating function changes as \(\beta\) changes. \(B\) describes the actual variation of that estimating function. Under misspecification, it is \(V_i\), not \(W_i\), that determines the variability.

Proof. At the true parameter, the marginal mean is correct, so \(E[Y_i-\mu_i(\beta_0)]=0\) and \(E[U_i(\beta_0)]=0\). Differentiating \(U_i\) with respect to \(\beta^\mathsf{T}\), the term involving derivatives of \(D_i\) is multiplied by the mean-zero residual and has expectation zero. The remaining expected derivative is \(-E[D_i^\mathsf{T}W_i^{-1}D_i]=-A\). The assumed regularity makes the average derivative converge to \(-A\) in probability.

A Taylor expansion of the average estimating function between \(\beta_0\) and \(\hat\beta\) therefore gives

Because clusters are independent, the cluster central limit theorem gives \(m^{-1/2}\sum_i U_i(\beta_0)\xrightarrow{d}N(0,B)\): the covariance of one cluster’s estimating function is

Rearranging the Taylor expansion and multiplying by \(\sqrt{m}\) yields \(\sqrt{m}(\hat\beta-\beta_0)=A^{-1}m^{-1/2}\sum_i U_i(\beta_0)+o_p(1)\). The central limit theorem and the continuous mapping theorem give the stated limit, whose covariance is \(A^{-1}BA^{-1}\). \(\square\)

Worked Two-Visit Comparisons

For the following examples, each cluster has two responses with unit marginal variances. The true covariance and the working covariance are

These are positive definite correlation matrices. Consider the correctly specified one-parameter marginal mean model \(E[Y_i]=\beta x\), where \(x=(1,2)^\mathsf{T}\). The example isolates how working weights combine the two visits.

When Correct Working Covariance Is Optimal

The preceding comparison is consistent with a useful optimality result: among linear unbiased estimating procedures for a scalar coefficient with the same cluster design vector, weighting by the true covariance has the smallest variance. This does not say that every misspecified working covariance is worse than every simpler one. It says the true covariance is optimal within this particular class.

Proof. Put \(h=q-q_*\). Both \(q\) and \(q_*\) satisfy the unbiasedness constraint, so \(h^\mathsf{T}d=0\). Since \(Vq_*=d/(d^\mathsf{T}V^{-1}d)\), it follows that \(h^\mathsf{T}Vq_*=0\). Expanding the variance gives

The inequality holds because \(V\) is positive definite. Also, direct substitution gives \(q_*^\mathsf{T}Vq_*=1/(d^\mathsf{T}V^{-1}d)\). Equality holds only when \(h=0\). \(\square\)

Practical Interpretation and a Common Mistake

The sandwich covariance separates the effect of the working model on sensitivity from the actual variability of the data. A working correlation that is wrong can still yield a consistent estimate when the marginal mean is correct and the estimating equation remains valid. But consistency does not imply that the estimate has the smallest possible variance.

• Common mistake: “The marginal mean is correct, so the model-based covariance must also be correct.” The correction is that the mean determines the target, while the true response covariance determines the variability of the estimating function. Under correlation misspecification, use \(B\) formed with \(V_i\), not a covariance obtained by simply treating \(W_i\) as the truth.
• Another mistake: “A more flexible working correlation must be more efficient.” The two-visit comparison shows why this is false: poor working weights can increase variance. Flexibility is useful only insofar as it represents dependence well for the estimation problem at hand.