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

JoonHo Lee

Abstract

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.

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

Abstract

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.
Paper Structure (175 sections, 54 theorems, 144 equations, 13 figures, 28 tables, 1 algorithm)

This paper contains 175 sections, 54 theorems, 144 equations, 13 figures, 28 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivating Data and Preliminary Analysis
  3. The NSECE survey
  4. Four data features and their modeling implications
  5. Structural zeros and the hurdle specification.
  6. Bounded support and the beta-binomial kernel.
  7. Overdispersion and the concentration parameter $\kappa$.
  8. Complex survey design and the pseudo-posterior.
  9. Preliminary evidence for the poverty reversal
  10. Pooled estimates.
  11. State heterogeneity.
  12. Policy correlations.
  13. The Hierarchical Hurdle Beta-Binomial Model
  14. Notation and data structure
  15. The hurdle beta-binomial likelihood
  16. ...and 160 more sections

Key Result

Theorem 1

For all $n \ge 2$, $\kappa > 0$, and $\mu \in (0,1)$, For $n = 1$, $h(\mu) \equiv 1$ for all $\mu \in (0,1)$.

Figures (13)

  • Figure 1: Distribution of the infant/toddler enrollment share $Y_i/n_i$ across $N = 6{,}785$ center-based providers (NSECE 2019). The spike at zero (35.3% of centers) represents providers that do not serve any children under age 3. Among IT-serving providers ($z_i = 1$, $N_{\mathrm{pos}} = 4{,}392$), the distribution is roughly unimodal with a mean of 0.478 and substantial dispersion (SD $= 0.214$), consistent with overdispersion approximately twelve times beyond the binomial---three features that jointly motivate the hurdle beta-binomial model.
  • Figure 2: Preliminary evidence for the poverty reversal in the raw NSECE 2019 data. Left panel: IT participation rate (fraction of centers serving any infants) by community poverty decile; a loess smoother with 95% confidence band confirms the declining trend. Right panel: mean IT enrollment share ($Y_i/n_i$) among servers by poverty decile, showing the opposing positive trend. The two panels together display the reversal in its simplest form: participation falls with poverty, but intensity rises.
  • Figure 3: Empirical coverage of 90% intervals across $R = 200$ replications. Each panel corresponds to a simulation scenario (S0, S3, S4 with increasing Kish $\text{DEFF}$). Points show coverage rates for each parameter--estimator combination; the horizontal dashed line marks the 90% nominal level; the shaded band indicates $\pm\,2$ MCSE ($\approx 4.2$ percentage points). E-UW (unweighted) maintains near-nominal coverage throughout. E-WT (weighted, naive) collapses as $\text{DEFF}$ increases. E-WS (sandwich-corrected) provides meaningful recovery for fixed effects but cannot correct the hyperparameters $\tau_{\mathrm{ext}}$ and $\tau_{\mathrm{int}}$.
  • Figure 4: Impact of the sandwich variance correction on fixed-effect inference. Left panel: naive MCMC 95% credible intervals (gray) versus sandwich-corrected Wald 95% confidence intervals (colored) for all 11 fixed effects. The naive intervals reflect prior width; the Wald intervals reflect data information adjusted for the survey design. Right panel: design effect ratio ($\operatorname{DER}$) for each parameter. A $\operatorname{DER}$ of 1 indicates no survey design effect; the observed range is 1.14--4.18 (mean 2.11).
  • Figure 5: Cross-margin scatter plot of state-specific poverty coefficients from the unweighted M2 model (block-diagonal SVC). Each point represents the posterior mean $(\tilde{\alpha}_{\mathrm{pov},s},\;\tilde{\beta}_{\mathrm{pov},s})$ for one state. Small points without labels denote states with $N < 40$. The four quadrants correspond to different poverty--enrollment patterns: the upper-left quadrant (shaded) contains 48 of 51 states (94%), indicating a near-universal reversal pattern under hierarchical shrinkage. The dashed lines indicate the population-average coefficients $(\alpha_{\mathrm{pov}},\;\beta_{\mathrm{pov}})$. The near-zero cross-margin correlation ($\hat{\varrho} \approx 0.021$) implies that the two margins respond to poverty through largely independent mechanisms.
  • ...and 8 more figures

Theorems & Definitions (149)

  • Definition 1: Hurdle beta-binomial model
  • Theorem 1: Monotonicity of $h$
  • proof : Proof sketch
  • Proposition 1: Elasticity of $h$
  • Proposition 2: Non-identification of $\boldsymbol{\Sigma}_{12}$ from the conditional likelihood
  • proof
  • Remark 1: Conditional versus marginal likelihood
  • Proposition 3: Identifiability of $\boldsymbol{\Sigma}_{12}$ via the prior
  • Remark 2: Boundary condition
  • Definition 2: Policy moderator specification
  • ...and 139 more