For target-weighted group average treatment effects under population shift, the optimal two-stage design allocates groups in proportion to sqrt(deployment share) times the sum of arm standard deviations, with the Neyman split within groups.
Optimalityofmatched-pairdesignsinrandom- izedcontrolledtrials.AmericanEconomicReview,112(12): 3911–3940
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Target-Weighted Neyman Allocation: Experimental Design for Heterogeneous Treatment Effects under Population Shift
For target-weighted group average treatment effects under population shift, the optimal two-stage design allocates groups in proportion to sqrt(deployment share) times the sum of arm standard deviations, with the Neyman split within groups.