Experimental Design for Target-Population Treatment Effect Estimation under Heterogeneous Privacy Constraints

Estimating a target-population average treatment effect (ATE) in multisite randomized trials requires allocating limited enrollment across sites that differ in target-population relevance, recruitment costs, and privacy constraints. Existing target-population trial designs do not directly account for site-specific privacy constraints, while private multisite ATE estimation generally takes site enrollment as given. In this paper, we study experimental design for multisite randomized trials under site-specific participant-level privacy. We characterize each site's privacy-constrained information contribution and derive a target-specific criterion that balances estimation variance and worst-case squared bias. The criterion characterizes the minimax mean squared error within constants independent of the budget, site count, and allocation. A site's information contribution grows quadratically with enrollment in the privacy-limited branch and linearly after a site-specific threshold, so heterogeneous privacy can change the allocation favored by classical design. The resulting design problem is strongly NP-hard even in the high-privacy regime. We develop sparse conic constructions with dimension-dependent approximation guarantees under explicit solver-accuracy conditions, budget-feasible integer recovery, and private estimator weights. Experiments show when accounting for heterogeneous privacy reduces target-population ATE error under a fixed enrollment budget and when the added privacy variance outweighs the reduction in squared bias.