Improving Rare Event Estimation in Public Health Surveys: A Bayesian Hierarchical Approach Within Adaptive Cluster Sampling

Accurate estimation of rare events in clustered populations is paramount for effective public health planning, yet traditional estimators often assume population homogeneity, leading to inefficiencies and potential bias. This study employs a case study approach using a real-world public health dataset to compare two estimators for the proportion of a rare event (incomplete Hepatitis B vaccination) within an Adaptive Cluster Sampling (ACS) frame-work: a classical ACS estimator and a heterogeneous estimator derived from a Bayesian hierarchical model. The hierarchical structure accounts for clustering at the College-Department level (networks, representing 279 distinct net-work units identified through the adaptive sampling process, including both expanded clusters and singleton networks) and the College-Year level (sub-networks, representing 45 nested groupings within the 380-student population). By treating a complete dataset as a known (population), we rigorously assess estimator performance in terms of bias, variance, and Mean Squared Error (MSE). Our findings demonstrate that the Bayesian hierarchical estimator (estimate: 0.182, variance: 0.00038, MSE: 0.00038) consistently yields estimates with negligible bias, lower variance, and a substantially lower MSE compared to the classical ACS estimator (estimate: 0.172, variance: 0.00050, MSE: 0.00059). This represents a 1.31-fold reduction in variance and a 1.55-fold reduction in MSE for the Bayesian approach. Posterior predictive checks further confirm the good fit of the Bayesian model to the observed data. This underscores the critical importance of explicitly accounting for population heterogeneity and employing robust model-based inference in adaptive sampling designs to generate reliable estimates for informed public health policy.