Deriving Implied Sampling Costs from Fixed-Budget Stratified Allocation

Abstract

Objectives: Classical stratified allocation rules optimize statistical precision without regard to per-unit sampling costs. This study inverts that optimization problem to recover a closed-form, per-unit cost structure implied by any fixed-budget allocation, establishing a reverse diagnostic framework for survey auditing. Method: The Lagrangian solution to cost-constrained Neyman allocation was inverted to yield a general inverse-cost expression, with equal, proportional, and Neyman weights substituted to produce rule-specific closed-form formulas. The framework was validated through four tiers: algebraic derivation, a five-stratum provincial illustration, a 5,000-replication Monte Carlo simulation, and an empirical post-hoc audit of a published survey (HILDA Wave 18, L = 5 strata). Findings: Each allocation rule implied a distinct, non-cost-neutral cost structure. Equal allocation produced a uniform implied cost of ₱11,549.25 across all strata regardless of population size or variability, while proportional and Neyman-weighted allocations produced implied costs ranging from ₱657.52 to ₱75,870.99 depending on stratum size and variability. Monte Carlo results based on 5,000 replications showed mean absolute rounding error ranging from 0.83 percent under equal allocation to 2.90 percent under Neyman allocation by standard deviation, with individual-stratum errors as high as 85.88 percent and error escalating from 1.45 percent to 12.22 percent as the number of strata increased from three to twenty. In the empirical audit, the framework's recovered implied cost diverged from the originally reported cost by more than 500 percent in four of five strata, compared with a stepwise versus full-precision computational gap of no more than approximately 3 percent in the same allocation. Under the cost-only reading of the allocation, this indicates that the divergence reflects the structure of the allocation rule rather than a rounding artifact. Novelty: This study introduces the first inverse-cost diagnostic framework for stratified sampling, converting conventional allocation rules, previously treated as statistically neutral defaults, into falsifiable, auditable per-unit cost assumptions, and demonstrates its use as a post-hoc auditing tool for allocations already in the field. Keywords: Stratified sampling, Neyman allocation, Survey cost, Inverse-cost framework, Sample allocation