A Bayesian Hierarchical Hurdle Beta-Binomial Model for Survey-Weighted Bounded Counts and Its Application to Childcare Enrollment

Bounded discrete proportions -- counts out of known totals -- present modeling challenges when data exhibit structural zeros, overdispersion, and hierarchical clustering. We develop a Bayesian hierarchical hurdle beta-binomial model with state-varying coefficients that addresses all four features. The framework makes three methodological contributions: (i) it studies cross-margin dependence via a cross-block covariance component and clarifies when and how this parameter is identified through the hierarchical layer rather than the conditional likelihood; (ii) it proposes a Cholesky-based sandwich variance calibration for pseudo-posterior inference under survey weights, guided by a parameter-specific design effect ratio diagnostic; and (iii) it introduces a log-scale marginal effect decomposition for hurdle models that translates regression coefficients into policy-relevant quantities. Applied to 6,785 childcare providers across 51 states from the 2019 National Survey of Early Care and Education, the model reveals a "poverty reversal": poverty reduces enrollment participation yet increases intensity among participants, with the extensive margin accounting for two-thirds of the total effect. Design-calibrated simulation shows that sandwich-corrected intervals substantially improve coverage, reaching 82--88.5% at the 90% nominal level for fixed effects. The R package hurdlebb implements all methods.