Generalized Estimating Equations

Definition

Generalized Estimating Equations (GEE) is a semiparametric method for regression analysis of clustered or longitudinal data in which observations within a cluster are correlated. Introduced by Liang and Zeger (1986), GEE extends the generalized linear model (GLM) estimating equation by replacing the independence working assumption with a parametric working correlation structure $R_i(\alpha)$. The regression coefficients $\beta$ are estimated via quasi-score equations and are consistent regardless of whether $R_i(\alpha)$ is correctly specified; a robust (sandwich) variance estimator corrects for misspecification of the correlation structure.

Key Ideas

• Working correlation matrix: The working covariance for cluster $i$ is $V_i = \phi\, A_i^{1/2} R_i(\alpha) A_i^{1/2}$, where $A_i = \text{diag}(\nu_{ij})$ contains the marginal variance functions, $\phi$ is a dispersion parameter, and $R_i(\alpha)$ is the working correlation (exchangeable, AR-1, unstructured, or independent).
• GEE estimating equations: $\sum_i D_i' V_i^{-1}(y_i - \mu_i) = 0$, where $D_i = \partial\mu_i/\partial\beta$. With independent working correlation this reduces to standard GLM; with other choices it exploits within-cluster correlation to gain efficiency.
• Robust sandwich variance: $\hat{V}(\hat\beta) = M^{-1} \hat{B} M^{-1}$ where $M = \sum_i D_i' V_i^{-1} D_i$ and $\hat{B} = \sum_i D_i' V_i^{-1}\hat{e}_i\hat{e}_i'V_i^{-1}D_i$ (residual outer products). Consistency of $\hat{V}$ requires only correct specification of the marginal mean model, not of the correlation structure.
• Marginal vs. cluster-specific coefficients: For nonlinear link functions (e.g., logit), the GEE $\beta$ has a population-average (marginal) interpretation, while random-effects coefficients $\beta^{re}$ have a cluster-specific (subject-specific) interpretation. The two differ in magnitude ($|\beta^{re}| > |\beta^{marg}|$ for logistic link); for linear links they coincide.
• GEE2 (Prentice 1988): Extension that jointly models the first moment (means) and second moment (pairwise correlations) via separate estimating equations for $\beta$ and $\alpha$; more efficient than standard GEE when correlation parameters are of substantive interest, but less robust.
• The four model families for clustered binary data (Pendergast et al. 1996): (1) naive/response-feature (ignores correlation), (2) conditionally specified (auto-logistic, Rosner beta-binomial, quadratic exponential), (3) marginal/GEE, (4) cluster-specific/random effects.

How It Works

GEE Algorithm (Liang-Zeger 1986)

Iterate between updating $\beta$ from the quasi-score equations and updating $\alpha$ from moment estimators of the within-cluster residual correlations:

1. Initialize $\hat\beta$ from standard GLM (independent working correlation).
2. Compute Pearson residuals $\hat{e}_{ij} = (y_{ij} - \hat\mu_{ij})/v_{ij}^{1/2}$.
3. Estimate $\hat\alpha$ from $\hat{e}_{ij}$ using method-of-moments (e.g., $\hat\alpha = \frac{1}{n^*}\sum_{i}\sum_{j<k}\hat{e}_{ij}\hat{e}_{ik}$ for exchangeable $R_i$).
4. Update $V_i = \phi\, A_i^{1/2} R_i(\hat\alpha) A_i^{1/2}$ and re-solve $\hat\beta$ from the score equations.
5. Repeat until convergence.

The sandwich variance $\hat{V}(\hat\beta)$ is computed at convergence. Under correct marginal mean specification and mild regularity conditions, $n^{1/2}(\hat\beta - \beta_0) \to \mathcal{N}(0, V_0)$ where $V_0$ is the sandwich limit; the robust estimator is consistent regardless of whether $R_i(\alpha)$ is correct.

Common Working Correlation Structures

• Independence: $R_i = I$ (standard GLM; consistent but inefficient when correlation exists).
• Exchangeable: $R_{i,jk} = \rho$ for $j \neq k$; appropriate when within-cluster units are exchangeable (e.g., family members, eyes of the same patient).
• AR-1 (first-order autoregressive): $R_{i,jk} = \rho^{|j-k|}$; appropriate for longitudinal data with time-ordered observations.
• Unstructured: $R_{i,jk}$ estimated freely for all $(j,k)$; requires large clusters, many parameters.

Conditional Inference

For exchangeable binary data, conditioning on the cluster total $y_{i+} = \sum_j y_{ij}$ eliminates nuisance cluster effects. The conditional likelihood $P(y_i | y_{i+})$ does not depend on random cluster effects under the quadratic exponential family or when the within-cluster association parameter $\alpha$ is the only cluster-specific parameter. This approach (Altham 1978; Liang-Zeger-Qaqish 1992) sacrifices efficiency but provides valid inference without specifying the marginal distribution fully.

Goodness of Fit

Standard likelihood-based goodness-of-fit (GOF) statistics (deviance, likelihood ratio) are unavailable for GEE because it specifies only the first two moments, not a full likelihood. Alternatives include: score tests for association parameters $\alpha$; permutation-based tests; quasi-likelihood information criteria (QIC, Pan 2001); residual-based diagnostics. This is the main disadvantage of GEE relative to full-likelihood methods (beta-binomial, random-effects logistic).

Why It Matters

• Provides consistent regression coefficient estimates for clustered binary or count data without specifying a full joint distribution — only the marginal mean model must be correct.
• The sandwich estimator is the standard reference for inference in clustered data in biostatistics and clinical trials; it underlies cluster-robust standard errors (SEs) in panel econometrics.
• The marginal/cluster-specific distinction is fundamental: population-average effects (GEE) answer "what is the average treatment effect?" while cluster-specific effects (random effects) answer "what is the effect for a specific individual?"
• Widely used in clinical trials with cluster randomization, ophthalmological studies (bilateral data), longitudinal surveys, and familial aggregation studies.

Open Questions

• Efficiency of GEE relative to correctly specified parametric models; choice of working correlation structure affects efficiency but not consistency.
• Small-cluster asymptotics: sandwich SE is biased in finite samples with few clusters; corrections (Mancl-DeRouen 2001; Pan-Wall 2002) improve finite-sample coverage.
• Model selection for the working correlation structure remains informal; QIC and related criteria provide one approach but are not widely validated.
• Bayesian GEE extensions (Schildcrout-Heagerty 2005) introduce prior information on $\alpha$ but lack full posterior coherence.

Related

• Binary Probit
• Bayesian Hierarchical Model
• Pendergast et al. (1996) A Survey of Methods for Analyzing Clustered Binary Response Data