Bias-Reduced GEE via Adjusted Estimating Equations, with Odds-Ratio Extensions

Technical Summary: Bias-Reduced GEE via Adjusted Estimating Equations, with Odds-Ratio Extensions

Problem Statement
Generalized Estimating Equations (GEE), introduced by Liang and Zeger [1986], are a standard approach for analyzing correlated data, such as longitudinal studies. While GEE estimators are consistent and asymptotically normal under mild regularity conditions, they can exhibit substantial finite-sample bias when the number of independent clusters is small to moderate. This bias affects Wald-type inference, leading to inflated Type I error rates.

Existing literature on bias reduction for GEE faces several limitations. First, methods by Paul and Zhang [2014] and Lunardon and Scharfstein [2017] often treat the working covariance matrices as fixed with respect to the mean parameters. This simplification fails to account for the dependence of the variance on the mean in binary and Poisson responses, preventing first-order bias reduction in these common settings. Second, for binary responses, existing bias-correction methods rely on correlation-coefficient parameterizations. These parameterizations are subject to restrictive feasibility constraints (Fréchet bounds) imposed by marginal means, which can cause correlation estimates to fall outside admissible ranges or render the working correlation matrix non-positive definite, particularly in small samples. Finally, existing bias-corrected estimators require a well-defined ordinary GEE estimator as a starting point, making them numerically unstable or inapplicable in cases of separation or near-separation.

Methodology
The paper develops a unified first-order bias-reduction framework by embedding the GEE estimator within the clustered-data M-estimation framework, building on the robust bias-reduction methodology of Kosmidis and Lunardon [2024].

1. Bias-Reducing Adjustment: The authors derive an explicit target adjustment, $\Lambda_G$, to the GEE estimating equations. This derivation treats the estimator as an M-estimator and explicitly accounts for the dependence of the working covariance matrices on the mean parameters. The ideal adjustment targets the leading bias term, $\Lambda_G$, which depends on unknown covariance matrices.
2. Three Implementable Estimators: To make the adjustment practical, three approximations are constructed:
   • Robust Adjustment (RBR): Replaces unknown covariance matrices with unbiased empirical analogs ($S_i S_i^\top$). This estimator achieves first-order bias reduction regardless of whether the working association structure is correctly specified.
   • Naive Adjustment (NBR): Assumes the working covariance equals the true covariance. This nests the estimators of Paul and Zhang [2014] as a special case (where the variance dependence on the mean is ignored). It achieves bias reduction if the association is correctly specified or under specific conditions even if misspecified.
   • Empirical Adjustment (EBR): Replaces all expectations with sample analogs. While theoretically valid, it may inflate finite-sample variability and is recommended only as a fallback when the robust procedure is numerically unstable.
3. One-Step Bias-Corrected Analogs: Computationally convenient one-step estimators are derived by applying the adjustment to the ordinary GEE estimate. These avoid iterating the adjusted equations but require the existence of the ordinary GEE solution.
4. Odds-Ratio Parameterization: The framework is extended to correlated binary data using pairwise odds-ratio parameterizations (Lipsitz et al., 1991; Carey et al., 1993). This approach avoids the restrictive feasibility constraints of correlation coefficients. The authors derive the necessary adjustments ($\Lambda_{OR}$) for this parameterization, yielding the first bias-reduced and bias-corrected GEE estimators under odds-ratio modeling.
5. Inference: For small-sample inference, the paper recommends using the small-sample adjusted covariance estimator of Morel et al. [2003] ($\hat{\Sigma}_M$) rather than the standard sandwich estimator.

Key Results
Simulation studies involving 10,000 Monte Carlo replications for correlated binary (probit model) and Poisson responses were conducted across various cluster sizes ($N \in \{20, 35, 50, 100, 500\}$) and missingness patterns (MCAR monotone dropout).

• Bias Reduction: The Robust Bias-Reduced (RBR) estimator consistently reduced finite-sample bias compared to ordinary GEE, particularly for time-varying covariates and in small-to-moderate sample sizes. The bias reduction was most pronounced for binary responses.
• Efficiency and Coverage: The RBR estimator maintained efficiency comparable to ordinary GEE. The empirical coverage of nominal 95% Wald confidence intervals based on $\hat{\Sigma}_M$ was close to the nominal level for RBR, whereas ordinary GEE often exhibited undercoverage in small samples.
• Robustness to Misspecification: The RBR estimator performed well even when the working association structure was misspecified. The Naive Bias-Reduced (NBR) estimator also showed bias reduction under misspecification in certain scenarios, confirming theoretical findings.
• Numerical Stability: The Empirical Bias-Reduced (EBR) estimator showed numerical instability in small samples with missing data, exhibiting lower convergence rates and inflated standard errors, validating the recommendation to use RBR as the default.
• Odds-Ratio vs. Correlation: In a clinical trial application (shoulder-tip pain data), fitting the model with an unstructured working correlation matrix led to numerical difficulties (non-convergence, inadmissible estimates). In contrast, the odds-ratio parameterization provided stable estimates, demonstrating its superiority for flexible modeling of binary associations in small samples.

Significance and Claims
The paper claims to provide the first unified framework for bias reduction in GEE that:

1. Explicitly accounts for the dependence of the working covariance on mean parameters, thereby guaranteeing first-order bias reduction for mean-dependent variance functions (binary/Poisson) where previous methods failed.
2. Extends bias reduction to odds-ratio parameterizations for correlated binary data, offering a stable alternative to correlation-based methods that suffer from admissibility constraints in small samples.
3. Subsumes and clarifies the relationship between existing methods (Paul and Zhang [2014]; Lunardon and Scharfstein [2017]), showing them as special cases of the proposed robust and naive frameworks.
4. Offers a practical solution for settings where ordinary GEE fails (e.g., separation) by providing fully iterated bias-reduced estimators that can be initialized via Firth's bias-reduced GLM.

The authors conclude that the Robust Bias-Reduced (RBR) estimator, implemented in the R package geer, should serve as the practical default for small-to-moderate sample sizes, particularly for correlated binary data, as it reduces bias while maintaining the asymptotic properties and efficiency of ordinary GEE.