E‐optimal designs for regression models with quantitative factors— a reasonable choice?

Abstract

AbstractFor regression models with quantitative factors it is illustrated that the E‐optimal design can be extremely inefficient in the sense that it degenerates to a design which takes all observations at only one point. This phenomenon is caused by the different size of the elements in the covariance matrix of the least‐squares estimator for the unknown parameters. For these reasons we propose to replace the E‐criterion by a corresponding standardized version. The advantage of this approach is demonstrated for the polynomial regression on a nonnegative interval, where the classical and standardized E‐optimal designs can be found explicitly. The described phenomena are not restricted to the E‐criterion but appear for nearly all optimality criteria proposed in the literature. Therefore standardization is recommended for optimal experimental design in regression models with quantitative factors. The optimal designs with respect to the new standardized criteria satisfy a similar invariance property as the famous D‐optimal designs, which allows an easy calculation of standardized optimal designs on many linearly transformed design spaces.