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

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.