Cluster-level analysis in three-level cluster randomised trials with multi-level covariate-dependent missingness

Community-based public health interventions often use three-level cluster randomised trials (CRTs), for example, individuals are nested within households which are again nested within communities. In these trials, outcome missingness commonly depends on baseline covariates measured at multiple levels. Although two-level CRTs are well studied, the effects of subcluster-level attrition and hierarchical covariate correlations on bias and variance are not yet fully explored. We develop an analytical framework for three-level CRTs under multi-level covariate-dependent missingness (CDM), explicitly incorporating within- and between-subcluster covariate correlations. We examine the performance of unadjusted and covariate-adjusted cluster-level treatment effect estimators using Monte Carlo simulations and a plasmode simulation based on the 2021 Madagascar Demographic and Health Survey. Our study reveals that unadjusted cluster-level analyses produce biased estimates under multi-level CDM. Linear mixed models are unbiased when correctly specified under missing-at-random assumptions but may be computationally unstable in small samples. The covariate-adjusted cluster-level estimator remains unbiased when covariate effects and missingness mechanisms are homogeneous across treatment arms. Subcluster-level attrition can lead to greater variance inflation than individual-level attrition. For CRTs with fewer than 30 clusters per arm and homogeneous treatment effects, covariate-adjusted cluster-level analysis may serve as a practical alternative to mixed models. A decision flowchart is provided to guide method selection.