Asymptotically optimal adaptive designs in factorial experiments

Abstract

Two-stage experiments with many two-level factors are considered under factorial designs. In the model, only main effects are present and responses are normal with equal variances. The purpose of experimentation is to choose factor settings that give a large expected response after experimentation. At the first stage, all factors are investigated. Using the information from the first stage experiment, factors with estimated factor effect near zero are included in the second stage for further investigation. Asymptotically optimal two-stage experiments are derived in a Bayesian decision theoretic formulation in which factors are i.i.d. with finite Fisher information. Both the number of runs and the number of factors tend to infinity in the limit considered. In the derivation, Stein's identity is used to approximate posterior risks and empirical process theory is used to deal with the large number of factors. The design obtained is near optimal for a wide class of prior distributions.