Saturation in G: simple & robust causal inference in cluster randomized trials with informative cluster sizes

Cluster randomized trials (CRTs) can exhibit informative cluster sizes (ICS) where cluster size is associated with outcomes and/or treatment effects. Under ICS, the individual and cluster-average treatment effects (iATE, cATE) can diverge, and the conventional linear mixed-effects model (LMM) and generalized estimating equation (GEE) with an exchangeable working correlation can produce data-dependent weighted contrasts that are not consistent for either estimand. In these settings with ICS, we propose easy to implement "cluster-size saturated models with g-computation" (CS-g), which employ a simple two-step adjustment to standard practice: (1.) augment the appropriately weighted working LMM or GEE with a saturated continuous cluster-size main effect and treatment x cluster-size interaction, and (2.) apply g-computation to target an interpretable marginal estimand. We prove that the appropriately weighted cluster-size saturated LMM with g-computation and more general cluster-size saturated GEE with g-computation can consistently target the iATE and cATE, among a broad class of interpretable estimands, while allowing for ICS. Crucially, this consistency holds under arbitrary misspecification of other model components, including the functional form of the saturated cluster-size terms. Furthermore, we demonstrate exact finite-sample equivalence between these consistent CS-g estimators and their model-robust standardization counterparts. Across simulations with continuous and binary outcomes, the proposed CS-g estimators were unbiased, more efficient than other consistent estimators, and returned greater power to detect ICS. A re-analysis of the PPACT P-CRT further illustrates the approach. Altogether, CS-g offers a simple, robust, and efficient route to target interpretable marginal effects in P-CRTs with ICS.