Optimal subsampling methods for large clustered imbalanced data based on generalized estimating equations

Generalized estimating equations (GEEs) are widely used for modeling clustered data while accounting for within-cluster correlations. However, fitting GEEs on massive datasets can be computationally prohibitive. Subsampling offers a practical solution, but efficient subsampling strategies for large clustered data, particularly with rare events, remain underexplored. Simple random subsampling can lead to a substantial loss of rare-event information, resulting in inefficient estimation, computational instability, and increased finite-sample bias. Additionally, intra-cluster correlation complicates inference on subsampled data, requiring methods that preserve correlation structure while adjusting for unequal sampling probabilities. To address these challenges, we develop an individual-level sampling scheme and derive optimal subsampling probabilities based on GEEs. We propose a subsampling-based weighted GEE (SaWGEE) estimator that efficiently approximates the full-data GEE estimator and establish its asymptotic properties as the number or size of clusters grows. The optimal subsampling probabilities minimize the asymptotic mean-square-error or its upper bound of the SaWGEE estimator, conditional on observed data, under an independent or exchangeable working correlation structure. Simulation studies evaluate the performance of the SaWGEE estimator against conventional subsampling strategies, such as random or 1:1 case-control sampling, and the standard full-data GEEs when feasible. We apply the proposed method to the 2022 CDC survey data, where observations are clustered by state, to construct a risk-adjustment model for heart attack incidence.