Covariate-adaptive treatment allocation is considered in the situation when a compromise must be made between information (about the dependency of the probability of success of each treatment upon influential covariates) and cost (in terms of number of subjects receiving the poorest treatment). Information is measured through a design criterion for parameter estimation, the cost is additive and is related to the success probabilities. Within the framework of approximate design theory, the determination of optimal allocations forms a compound design problem. We show that when the covariates are i.i.d. with a probability measure $\mu$, its solution possesses some similarities with the construction of optimal design measures bounded by $\mu$. We characterize optimal designs through an equivalence theorem and construct a covariate-adaptive sequential allocation strategy that converges to the optimum. Our new optimal designs can be used as benchmarks for other, more usual, allocation methods. A response-adaptive implementation is possible for practical applications with unknown model parameters. Several illustrative examples are provided.
"Information-regret compromise in covariate-adaptive treatment allocation." Ann. Statist. 45 (5) 2046 - 2073, October 2017. https://doi.org/10.1214/16-AOS1518