Optimal Supersaturated Designs

Abstract

We consider screening experiments where an investigator wishes to study many factors using fewer observations. Our focus is on experiments with two-level factors and a main effects model with intercept. Since the number of parameters is larger than the number of observations, traditional methods of inference and design are unavailable. In 1959, Box suggested the use of supersaturated designs and in 1962, Booth and Cox introduced measures for efficiency of these designs including E(s2), which is the average of squares of the off-diagonal entries of the information matrix, ignoring the intercept. For a design to be E(s2)-optimal, the main effect of every factor must be orthogonal to the intercept (factors are balanced), and among all designs that satisfy this condition, it should minimize E(s2). This is a natural approach since it identifies the most nearly orthogonal design, and orthogonal designs enjoy many desirable properties including efficient parameter estimation. Factor balance in an E(s2)-optimal design has the consequence that the intercept is the most precisely estimated parameter. We introduce and study UE(s2)-optimality, which is essentially the same as E(s2)-optimality, except that we do not insist on factor balance. We also provide a method of construction. We introduce a second criterion from a traditional design optimality theory viewpoint. We use minimization of bias as our estimation criterion, and minimization of the variance of the minimum bias estimator as the design optimality criterion. Using D-optimality as the specific design optimality criterion, we introduce D-optimal supersaturated designs. We show that D-optimal supersaturated designs can be constructed from D-optimal chemical balance weighing designs obtained by Galil and Kiefer (1980, 1982), Cheng (1980) and other authors. It turns out that, except when the number of observations and the number of factors are in a certain range, an UE(s2)-optimal design is also a D-optimal supersaturated design. Moreover, these designs have an interesting connection to Bayes optimal designs. When the prior variance is large enough, a D-optimal supersaturated design is Bayes D-optimal and when the prior variance is small enough, an UE(s2)-optimal design is Bayes D-optimal. While E(s2)-optimal designs yield precise intercept estimates, our study indicates that UE(s2)-optimal designs generally produce more efficient estimates for the main effects of the factors. Based on theoretical properties and the study of examples, we recommend UE(s2)-optimal designs for screening experiments.