Optimal Biased Weighing Designs and Two-Level Main-Effect Plans
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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.
KeywordsBaseline parameterization Chemical balance weighing design Φp-optimality Spring balance weighing design Type 1 optimality
AMS Subject Classification62K05
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- Ceranka, B., and K. Katulska. 1992. Relations between optimum biased spring balance weighing designs and optimum chemical balance weighing designs. Proc. 10th Conf. Probab. Math. Statist., Bratislava, Czechoslovakia, 95–101.Google Scholar
- Kiefer, J. 1975. Optimality and construction of generalized Youden designs. In A survey of statistical designs and linear models, ed. J. N. Srivastava, 333–353. Amsterdam, The Netherlands: North-Holland.Google Scholar
- Pukelsheim, F. 1989. Complete class results for linear regression designs over the multi-dimensional cube. In Contributions to probability and statistics. Essays in honor of Ingram Olkin, ed. L. J. Gleser, M. D. Perlman, S. J. Press, and A. R. Sampson, 349–356, New York, Springer.CrossRefGoogle Scholar
- Yates, F. 1935. Complex experiments. J. R. Stat. Soc. Ser. B, 2, 181–223.Google Scholar