More with Less - Bethel Allocation and Precision-Preserving Sample Size Reduction via Hierarchical Bayes Modelling

TL;DR

This paper introduces a two-stage strategy combining Bethel allocation and Hierarchical Bayes modelling to minimize sample size while ensuring precision across multiple variables and domains, addressing a key challenge for statistical offices.

Contribution

It presents a novel combination of Bethel allocation and Hierarchical Bayes models to optimize sample size reduction while maintaining precision constraints across multiple variables and regions.

Findings

The approach effectively reduces sample size while meeting precision targets.

Validation via Monte Carlo shows accurate and reliable estimates.

Method outperforms traditional ad hoc allocation techniques.

Abstract

Statistical offices face a familiar and intensifying dilemma: rising demand for detailed regional and domain-level estimates under budgets that are fixed or shrinking. National statistical offices (NSOs) either ignore the problem of optimal sample allocation for multiple target variables when designing a multi-purpose survey, or address it incorrectly - relying on ad hoc approaches such as computing Neyman allocations separately per variable and taking the element-wise maximum, a practice that simultaneously wastes budget and fails to guarantee precision across all domains. This paper presents a practical two-stage strategy that reframes the question: not how to allocate a given sample, but how small the sample can be made while still meeting pre-defined precision targets for all target variables across all geographic domains at once. The innovation lies not in inventing new methods, but in the novel combination of two well-established techniques applied to this cost-reduction problem: (i) multivariate constrained optimisation via Bethel allocation, which finds the globally minimum sample satisfying all precision constraints simultaneously; and (ii) Hierarchical Bayes (HB) small area modelling, which borrows strength across strata and permits a further reduction of the Bethel sample. The approach is validated using a Monte Carlo study (B = 1,000 replications) based on a synthetic labour-force population of one million individuals, where known population truth allows rigorous evaluation of precision, accuracy, and credible-interval coverage. Keywords: Bethel allocation; Hierarchical Bayes; small area estimation; sample size reduction; multivariate optimisation; labour force survey; coefficient of variation.