Asymptotically optimal adaptive designs in factorial experiments for quality improvement.

Abstract

Optimal designs are considered with many binary factors under two models. In the first model, only main effects are considered and responses are normal with equal variances. The design objective is to determine factor settings that give a large expected response after experimentation. In the second model, responses are still normal, but now the variance depends on factor settings. This model is appropriate for the parameter design approach to quality improvement. The design objective is to determine factor settings to minimize response variation. Asymptotically optimal two-stage experiments are obtained in a Bayesian decision theoretic formulation in which factors are iid with finite Fisher information and absolutely continuous distribution. 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 study. 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 designs obtained are robust to a wide class of prior distributions. The estimation and two-stage procedures for the first model are adapted for the second model. A formal account of the asymptotic performance of the resulting procedure is not attempted, but we suspect the procedure is near optimal provided that there are a fair number of replications. The performance of the various designs is studied numerically in situations where the number of factors and run sizes are moderate. The designs based on asymptotic theory perform well.