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

  • Ching-Shui ChengEmail author


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 Φp-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 entries, and the designs can also be considered as biased chemical balance weighing designs. On the other hand, under a nonorthogonal baseline parameterization, the model matrices have 0, 1 entries, and the designs are equivalent to biased spring balance weighing designs.


Baseline parameterization Chemical balance weighing design Φp-optimality Spring balance weighing design Type 1 optimality 

AMS Subject Classification



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Copyright information

© Grace Scientific Publishing 2014

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan

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