Statistical power and optimum sample allocation ratio for treatment and control having unequal costs per unit of randomization

This article considers optimal sample allocation between the treatment and control condition in multilevel designs when the costs per sampling unit vary due to treatment assignment. Optimal unequal allocation may reduce the cost from that of a balanced design without sacrificing any power. The optimum sample allocation ratio depends only on the cost ratio between the treatment and control regardless of whether the randomization of sampling units occurs at levels 1, 2, or 3. Power functions for the exact tests for the main effect of treatment are derived for prototypical multilevel designs with unequal sample sizes in the treatment and control condition.