Confidence Intervals for GEE Contrasts

Why Confidence Intervals for Contrasts Need Their Own Standard Errors

A GEE model may contain several coefficients, but scientific questions are often about a combination of them. For example, a treatment effect at a particular visit may combine a treatment coefficient with a treatment-by-visit interaction. The uncertainty in that combination depends not only on the uncertainty in each coefficient, but also on their covariance.

The robust covariance estimator used for GEE inference accounts for within-cluster dependence under the usual large-number-of-independent-clusters justification. For a contrast, the relevant uncertainty is obtained by applying that covariance estimator to the contrast itself. It is not generally correct to use the standard error of one coefficient, or to combine coefficient standard errors as if the estimates were independent.

Here, \(c\) is fixed by the scientific question and the model parameterization. A zero entry means the corresponding coefficient does not contribute to the contrast. The off-diagonal entries of \(\hat V\) matter whenever two or more coefficients contribute.

Large-Sample Confidence Interval for a Linear Contrast

State the asymptotic conditions in terms of the number \(K\) of independent clusters. Suppose the true coefficient vector is an interior point of the parameter space, the GEE estimator is consistent, and \(\sqrt{K}(\hat\beta-\beta)\xrightarrow{d}N(0,\Sigma)\), where \(\Sigma\) is finite and positive definite. Suppose also that the robust covariance estimator is consistent on this scale, so \(K\hat V\xrightarrow{p}\Sigma\). Standard regularity conditions for GEE are used to justify these limits; the clusters, rather than individual repeated observations within a cluster, supply the independent units.

Proof. Because \(c\) is fixed, applying a linear transformation to the assumed asymptotic normal limit gives \(\sqrt{K}(\hat\psi-\psi)\xrightarrow{d}N(0,c^\top\Sigma c)\). Also, \(Kc^\top\hat Vc\xrightarrow{p}c^\top\Sigma c\). Since this limiting variance is positive, taking square roots and using continuity gives \(\sqrt{K}\sqrt{c^\top\hat Vc}\xrightarrow{p}\sqrt{c^\top\Sigma c}\). Slutsky’s theorem therefore yields \[ \frac{\hat\psi-\psi}{\sqrt{c^\top\hat Vc}}\xrightarrow{d}N(0,1). \] The probability that this statistic lies between \(-z_{1-\alpha/2}\) and \(z_{1-\alpha/2}\) converges to \(1-\alpha\). Rearranging that event gives the stated interval. \(\square\)

For a 95% interval, \(z_{0.975}\) is approximately \(1.96\). The interval is a large-sample, pointwise interval for the specified contrast. Its validity depends on the asymptotic conditions, not on the working correlation being exactly correct: the robust covariance estimator is intended to provide valid large-sample uncertainty under working-correlation misspecification when the mean model and other required conditions are adequate.

Interpreting Contrasts on the Link Scale

In a GEE with a logit link, the modeled marginal mean \(\mu\) satisfies \(\operatorname{logit}(\mu)=\log\{\mu/(1-\mu)\}=x^\top\beta\). A linear contrast in the coefficients can therefore represent a difference in marginal log odds. Exponentiating that contrast gives a marginal odds ratio, but the interval should first be constructed on the log-odds scale.

Nonlinear Contrasts and the Delta Method

Some questions concern a smooth function \(g(\beta)\) rather than a linear combination. For example, a difference between two fitted marginal probabilities under a logit link is generally nonlinear in the coefficients. The delta method uses the local gradient of \(g\) to translate coefficient uncertainty into uncertainty for the target.

Derivation. Differentiability at \(\beta\) gives \(g(\hat\beta)-g(\beta)=d^\top(\hat\beta-\beta)+r\), where \(r=o(\|\hat\beta-\beta\|)\) as \(\hat\beta\to\beta\). Consistency and the asymptotic normal limit imply \(\hat\beta-\beta=O_p(K^{-1/2})\), so \(\sqrt{K}r=o_p(1)\). Consequently, \(\sqrt{K}\{g(\hat\beta)-g(\beta)\}\xrightarrow{d}N(0,d^\top\Sigma d)\). Continuity of the gradient at \(\beta\) and consistency of \(\hat\beta\) give \(\nabla g(\hat\beta)\xrightarrow{p}d\). Together with \(K\hat V\xrightarrow{p}\Sigma\), this implies \(K\nabla g(\hat\beta)^\top\hat V\nabla g(\hat\beta)\xrightarrow{p}d^\top\Sigma d\). Slutsky’s theorem gives the standardized normal limit, and rearrangement gives the interval. \(\square\)

Common Mistakes and Practical Interpretation

• Using one coefficient’s standard error for a combination. A contrast such as \(\hat\beta_T+5\hat\beta_{Ta}\) has variance involving both coefficient variances and their covariance. Use \(c^\top\hat Vc\).
• Ignoring covariance. The covariance may increase or decrease the contrast variance. It is not safe to assume independence unless it is justified by the covariance estimate.
• Building an odds-ratio interval on the wrong scale. Construct the interval for the log-odds contrast, then exponentiate its endpoints. Do not exponentiate the standard error or treat the coefficient-scale interval as already an odds-ratio interval.
• Treating a delta-method interval as exact. A nonlinear contrast interval is an approximation based on local linearization and large-sample behavior.
• Assuming differentiability alone validates a plug-in gradient. For the displayed delta-method interval, continuity of the gradient at the true parameter (or an explicit assumption that the estimated gradient is consistent) is needed.

The contrast should be specified before calculating its uncertainty: identify the target scale, express the target in terms of the model coefficients, and form the corresponding contrast vector or gradient. Interpret the resulting interval in the units of that target. As with other sandwich-based GEE intervals, the large-sample justification relies on having enough independent clusters; the normal approximation is not a guarantee of accurate coverage with few clusters.