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Coarsened data in small area estimation: a Bayesian two-part model for mapping smoking behaviour

Aldo Gardini, Lorenzo Mori

TL;DR

This work tackles the challenge of estimating health indicators from coarsened self‑reported data in small areas by introducing a Bayesian unit‑level two‑part SAE that jointly models prevalence and intensity while explicitly modeling coarsening and top‑coding. Using EHIS‑Italy data, it demonstrates that ignoring coarsening yields biased, unstable domain estimates, whereas the proposed LNM‑C model yields improved accuracy and near‑nominal coverage across a range of scenarios. Through simulation and an empirical application, the approach provides stable, interpretable region–age domain estimates for smoking indicators, enabling targeted public health decisions. The framework is extendable to other coarsened outcomes and is accompanied by Stan code to facilitate practical deployment.

Abstract

Estimating health indicators for restricted sub-populations is a recurring challenge in epidemiology and public health. When survey data are used, Small Area Estimation (SAE) methods can improve precision by borrowing strength across domains. In many applications, however, outcomes are self-reported and affected by coarsening mechanisms, such as rounding and digit preference, that reduce data resolution and may bias inference. This paper addresses both issues by developing a Bayesian unit-level SAE framework for semi-continuous, coarsened responses. Motivated by the 2019 Italian European Health Interview Survey, we estimate smoking indicators for domains defined by the cross-classification of Italian regions and age groups, capturing both smoking prevalence and intensity. The model adopts a two-part structure: a logistic component for smoking prevalence and a flexible mixture of Lognormal distributions for average cigarette consumption, coupled with an explicit model for coarsening and topcoding. Simulation studies show that ignoring coarsening can yield biased and unstable domain estimates with poor interval coverage, whereas the proposed model improves accuracy and achieves near-nominal coverage. The empirical application provides a detailed picture of smoking patterns across region-age domains, helping to characterize the dynamics of the phenomenon and inform targeted public health policies.

Coarsened data in small area estimation: a Bayesian two-part model for mapping smoking behaviour

TL;DR

This work tackles the challenge of estimating health indicators from coarsened self‑reported data in small areas by introducing a Bayesian unit‑level two‑part SAE that jointly models prevalence and intensity while explicitly modeling coarsening and top‑coding. Using EHIS‑Italy data, it demonstrates that ignoring coarsening yields biased, unstable domain estimates, whereas the proposed LNM‑C model yields improved accuracy and near‑nominal coverage across a range of scenarios. Through simulation and an empirical application, the approach provides stable, interpretable region–age domain estimates for smoking indicators, enabling targeted public health decisions. The framework is extendable to other coarsened outcomes and is accompanied by Stan code to facilitate practical deployment.

Abstract

Estimating health indicators for restricted sub-populations is a recurring challenge in epidemiology and public health. When survey data are used, Small Area Estimation (SAE) methods can improve precision by borrowing strength across domains. In many applications, however, outcomes are self-reported and affected by coarsening mechanisms, such as rounding and digit preference, that reduce data resolution and may bias inference. This paper addresses both issues by developing a Bayesian unit-level SAE framework for semi-continuous, coarsened responses. Motivated by the 2019 Italian European Health Interview Survey, we estimate smoking indicators for domains defined by the cross-classification of Italian regions and age groups, capturing both smoking prevalence and intensity. The model adopts a two-part structure: a logistic component for smoking prevalence and a flexible mixture of Lognormal distributions for average cigarette consumption, coupled with an explicit model for coarsening and topcoding. Simulation studies show that ignoring coarsening can yield biased and unstable domain estimates with poor interval coverage, whereas the proposed model improves accuracy and achieves near-nominal coverage. The empirical application provides a detailed picture of smoking patterns across region-age domains, helping to characterize the dynamics of the phenomenon and inform targeted public health policies.
Paper Structure (14 sections, 2 theorems, 31 equations, 7 figures, 2 tables)

This paper contains 14 sections, 2 theorems, 31 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The data and the motivation
  3. Notation and target quantities
  4. A two-part small area model for heaped data
  5. Part one: daily smoker proportion
  6. Part two: smoking intensity
  7. Model fitting, posterior inference and model checking
  8. Hierarchical Bayes predictors of target quantities
  9. Prior distributions
  10. Simulation study
  11. Smoking behaviour in Italy
  12. Model Selection and Validation
  13. Small Area Estimates
  14. Concluding remarks

Key Result

Proposition 1

Let us consider the likelihood accounting for heaping in eq:lik_real and the LNM distributional assumption in eq:distr_LNM. Gaussian priors with known parameters are set for $\beta_{01}^\mu$, $\beta_{02}^\mu$, and $\boldsymbol{\beta}^\mu$. Moreover, the random effects are assigned a Gaussian prior,

Figures (7)

  • Figure 1: Bar-plots of Daily smoker status (left panel) and distribution of average number of cigarettes smoked by daily smokers (right panel).
  • Figure 2: Domain-specific measures of performance for model-based estimators of $\overline{z}_d$ and $\overline{HS}_d$
  • Figure 3: Posterior predictive checks. Left-plot: Simulated predictive CDFs of $Z$ under different models, compared with the empirical CDF of $Z^\star$ (black line). Right-plot: Frequency distribution of $Z^\star$ (gray bars) compared with the distribution of predicted counts under LN-C and LNM-C models.
  • Figure 4: Relationship between the heaping level probabilities and the average number of cigarettes per day ($Z$).
  • Figure 5: Comparison of design- and model-based estimates and their standard errors.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof