Wald Inference for GEE Coefficients

From a Robust Standard Error to an Inference

A robust standard error is useful because it quantifies uncertainty in a fitted GEE coefficient without requiring the working correlation to be exactly correct. Wald inference turns that uncertainty into a test or interval. The essential comparison is between the estimated coefficient and the value specified by a null hypothesis, measured in robust standard-error units.

Here \(n\) denotes the number of independent clusters, and \(\beta=(\beta_1,\ldots,\beta_p)^\mathsf{T}\) is the vector of coefficients in the marginal mean model. Write \(\hat\beta\) for the GEE estimator and \(\widehat V_R\) for its robust, or sandwich, covariance estimate. The robust standard error of coefficient \(j\) is \(\widehat{\operatorname{SE}}_R(\hat\beta_j)=\sqrt{(\widehat V_R)_{jj}}\). This tutorial uses that standard error for inference about one coefficient at a time.

The robust covariance estimate is a matrix: its diagonal entries estimate the variances of the individual coefficient estimates, while its off-diagonal entries estimate covariances between them. For this one-coefficient Wald statistic, the relevant entry is the \(j\)th diagonal entry. Off-diagonal entries are not needed to test that coefficient alone.

Tests and Confidence Intervals

For a two-sided test at level \(\alpha\), reject \(H_0:\beta_j=\beta_{j,0}\) when \(|Z_j|>z_{1-\alpha/2}\), where \(z_{1-\alpha/2}\) is the corresponding standard-normal quantile. Equivalently, compare \(W_j\) with the \(1-\alpha\) quantile of a chi-squared distribution with one degree of freedom. The approximate two-sided p-value is \(2\{1-\Phi(|Z_j|)\}\), with \(\Phi\) the standard-normal cumulative distribution function.

A two-sided \(100(1-\alpha)\%\) Wald confidence interval for \(\beta_j\) is the estimate plus or minus the critical normal quantile times its robust standard error. It describes uncertainty on the coefficient scale, which is the scale of the linear predictor. The interval and test are connected: for a test of \(\beta_j=\beta_{j,0}\), the null value is excluded from this interval exactly when the corresponding two-sided Wald test rejects, using the same critical value.

The normal and chi-squared reference distributions are large-sample approximations, not automatic finite-sample laws. In the usual GEE asymptotic framework, the number of independent clusters increases, the marginal mean model is correctly specified, the relevant limiting covariance is nonsingular, and suitable moment and regularity conditions hold. The robust covariance estimator must consistently estimate that limiting covariance. In particular, many observations within a small number of clusters do not by themselves provide the same asymptotic justification as many independent clusters.

Why the Link Scale Matters

A GEE specifies a marginal mean through a link: \(g(\mu_{it})=x_{it}^{\mathsf T}\beta\), where \(\mu_{it}=E[Y_{it}\mid x_{it}]\) is a modeled marginal mean and \(g\) is the link function. A coefficient is a change in the linear predictor for a one-unit change in its covariate, with other covariates held fixed. Its response-scale meaning depends on \(g\).

With an identity link, a coefficient is a difference in marginal means. With a log link, exponentiating a coefficient gives a mean ratio for a one-unit covariate change. With a logit link, exponentiating a coefficient gives an odds ratio for the modeled marginal probabilities. These interpretations concern the specified marginal mean model; they are not automatically subject-specific effects.

A confidence interval for a coefficient can be transformed when the target is a monotone function of that coefficient. For instance, with a log link, exponentiate both endpoints of a coefficient interval to obtain an interval for the corresponding mean ratio. Because the exponential function is increasing, the endpoint order is preserved. This procedure is different from forming an interval for a fitted mean that depends on several coefficients; that target requires accounting for their covariance.

Inference for a Fitted Marginal Mean

Sometimes the scientific target is a fitted marginal mean at a specified covariate vector \(x\), rather than a coefficient itself. Define \(q(\beta)=h(x^\mathsf{T}\beta)\), where \(h=g^{-1}\) is the inverse link. A confidence interval for one coefficient cannot generally be turned into an interval for \(q(\beta)\): the fitted mean can depend on several coefficients, and their estimated covariances affect its uncertainty.

To justify this result, differentiability gives the first-order expansion \(q(\hat\beta)-q(\beta)=\nabla q(\beta)^\mathsf{T}(\hat\beta-\beta)+r_n\), where \(r_n=o_p(\|\hat\beta-\beta\|)\). The assumed asymptotic normality implies \(\hat\beta-\beta=O_p(n^{-1/2})\), so \(\sqrt n r_n\xrightarrow{p}0\). Therefore the first-order term determines the limiting distribution. Its variance is \(\nabla q(\beta)^\mathsf{T}\Sigma\nabla q(\beta)\). Consistency of \(\hat\beta\) and continuity of the gradient allow \(\nabla q(\beta)\) to be replaced by \(\nabla q(\hat\beta)\); consistency of \(n\widehat V_R\) supplies the corresponding estimated variance. The normal interval follows by standardizing this asymptotic distribution. \(\square\)

For \(q(\beta)=h(x^\mathsf{T}\beta)\), the gradient is \(h'(x^\mathsf{T}\beta)x\). Consequently the estimated variance is approximately \([h'(x^\mathsf{T}\hat\beta)]^2x^\mathsf{T}\widehat V_Rx\). For a logit link, \(h(\eta)=1/(1+\exp(-\eta))\) and \(h'(\eta)=h(\eta)\{1-h(\eta)\}\). The vector \(x\) makes clear why covariance terms between coefficients can matter for a fitted mean.

Deriving the Large-Sample Wald Reference

The Wald reference distribution follows from asymptotic normality and consistency of the robust covariance estimate. This argument also explains why robust standard errors, rather than working-model standard errors, enter the statistic when the working correlation may be misspecified.

Under the null, the assumed joint limit implies \(\sqrt n(\hat\beta_j-\beta_{j,0})\xrightarrow{d}N(0,\Sigma_{jj})\). Also, \(n(\widehat V_R)_{jj}\xrightarrow{p}\Sigma_{jj}\), so \(\sqrt{n(\widehat V_R)_{jj}}\xrightarrow{p}\sqrt{\Sigma_{jj}}\), which is positive by assumption. Slutsky’s theorem then gives

Squaring is a continuous transformation, so the continuous mapping theorem gives \(W_j=Z_j^2\xrightarrow{d}\chi^2_1\). This is a limiting result: it justifies approximate reference distributions as the number of independent clusters grows, not an exact finite-sample distribution. \(\square\)

Common Mistake and Practical Interpretation

• Mistake: “The working correlation is only an approximation, so the Wald test must be invalid.” Correction: With the stated regularity conditions, the robust covariance estimate accounts asymptotically for that misspecification in coefficient uncertainty. The Wald test is still an approximation whose quality depends on the independent-cluster asymptotics and covariance estimation.
• Mistake: “A coefficient interval is already an interval for the fitted mean.” Correction: A fitted mean may depend on several coefficients. Use the gradient and the full robust covariance matrix for its delta-method standard error.
• Mistake: “A log-link coefficient of \(0.12\) means a 12% increase exactly.” Correction: The corresponding ratio is \(\exp(0.12)\), about \(1.1275\), or a 12.75% increase. The coefficient itself is on the log-ratio scale.

Individual Wald inference is most useful when the scientific question concerns one named model coefficient, such as whether a treatment coefficient is zero. State the null value, the robust standard error, the reference distribution, and the scale of interpretation. Do not treat a small p-value as a measure of effect size: report the estimate and an interval alongside the test. If there are few independent clusters, ordinary large-sample Wald approximations can be unreliable; the number of rows within clusters does not remove that concern.