Optimal Biased Weighing Designs and Two-Level Main-Effect Plans

TLDR: In this paper, the authors considered the problem of optimal biased chemical and spring balance weights for the case N ≡ 0 (mod 4), where N is the run size, and showed that for all p ≥ 0, there is a Φ p -optimal estimator for this problem.

Abstract: Optimal biased chemical and spring balance weighing designs are considered. Optimal designs in either setting can be obtained from those in the other via a simple transformation. Optimal approximate designs for unbiased chemical balance, biased chemical balance, and biased spring balance are closely related and can easily be obtained from one another. These designs correspond to universally optimal exact designs for the case N ≡ 0 (mod 4), where N is the run size. While Cheng's (1980) result on the type 1 optimality of certain unbiased chemical balance weighing designs for the case N ≡ 1 (mod 4) can be extended to the biased setting, such an extension does not hold for N ≡ 2 (mod 4). We obtain exact Φ p -optimal designs in the latter case for all p ≥ 0. The results obtained in this article can also be applied to optimal main-effect plans when one is interested in the main effects but not the general mean. Under the usual orthogonal parameterization, the model matrices of main-effect plans have 1, −1 entri...