A -Optimal Factorial Designs for Test Versus Control Comparisons
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A‑Optimal Factorial Designs for Test Versus Control Comparisons Feng‑Shun Chai1 · Ashish Das2
© Grace Scientific Publishing 2020
Abstract The majority of the existing work on factorial designs centers around the wellknown orthogonal parametrization. In particular, main-effect plans compare the treatment levels of each factor by means of orthogonal parametrization. In contrast, significant research has been carried out for single-factor experiments for comparing a set of test treatments with a control treatment. Here, the comparison leads to a nonorthogonal parametrization. A-optimal designs for single-factor test-control experiments have been investigated in the literature. In practical situations, however, it is quite common that several factors are under study simultaneously. We study optimal designs for factorial experiments where test-control comparisons are of interest. We adopt a technique for finding A-efficient designs under the test-control parametrization via the approximate theory. We illustrate the procedure for finding A-efficient exact designs through few examples. Keywords Algorithm · Approximate theory · A-Efficiency · Nonorthogonal parametrization
1 Introduction Significant research has been carried out for single-factor experiments for comparing a set of test treatments with a control treatment. Such a problem arises, for example, in screening experiments or at the beginning of a long-term experimental investigation where it is initially desired to determine the relative performance of the new test treatments with respect to the control treatment. Here, the
* Ashish Das [email protected] Feng‑Shun Chai [email protected] 1
Academia Sinica, Taipei, Taiwan
2
Indian Institute of Technology Bombay, Mumbai, India
13
Vol.:(0123456789)
61
Page 2 of 8
Journal of Statistical Theory and Practice
(2020) 14:61
comparison leads to a nonorthogonal parametrization. For a review of single-factor designs under this setup, see Hedayat et al. [5] and Majumdar [6]. In many experiments, several factors are jointly responsible for the observed response. We consider multifactor experiments, where each factor has two or more levels, of which one is a control level. Our objective is in the comparisons of the control level of each factor with its remaining levels, called the test levels. As mentioned in Gupta [4], for illustration, consider the spraying trial experiment against botrytis in strawberries described in Pearce [8] involving two factors, the spraying substance and the spraying time. Each factor has three levels of which a control level is the level currently in use. The other two new levels of each factor are the test levels. The experiment compares (1) the standard preparation, the control level, with two new preparations, and (2) the usual spraying time, the control level, with two new spraying times. In another experiment involving four levels of mother’s strains and four levels of father’s strains of mice discussed in Armitage [1], one of the strains is the control level for
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