Construction of Generalized Minimum Aberration Three-Level Orthogonal Arrays with Three, Four and Five Columns
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Construction of Generalized Minimum Aberration Three‑Level Orthogonal Arrays with Three, Four and Five Columns Haralambos Evangelaras1 · Christos Peveretos1
© Grace Scientific Publishing 2020
Abstract In this paper, we construct generalized minimum aberration (GMA) three-level orthogonal arrays with three and four columns for any run size n ≡ 0 (mod 9) and GMA three-level orthogonal arrays with five columns for n ≡ 0, 18, 36, 81, 162, 207 and 225 (mod 243) runs. Efficient three-level orthogonal arrays with five columns are also constructed for n ≡ 99, 117, 180 and 198 (mod 243) runs. All constructed GMA designs produce GMA designs when projected onto lower dimensions. Moreover, they possess a minimum confounding structure when the linear-quadratic system of contrasts is used for modeling the response. Keywords Orthogonal arrays · Generalized minimum aberration · Generalized wordlength pattern · Linear-quadratic system of contrasts
1 Introduction In most experiments that examine m ≥ 3 factors, to evaluate the effects that the factors under investigation have on a response of interest, several values of the response variable are measured using a carefully determined design matrix. The rows of this matrix (that are also called runs in experimental design terminology) are selected out of the set of the different combinations of the levels of the factors. If all the different combinations of the levels of the factors are selected the same number of times r, a full factorial design with r replications is generated. If r = 1 , then the full factorial is said to be unreplicated. Full factorial designs guarantee that all factorial effects are estimated independently from each other and with the same precision. When the design is replicated, an independent estimation of the error variance can also be attained. * Haralambos Evangelaras [email protected] 1
Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece
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Vol.:(0123456789)
60
Page 2 of 17
Journal of Statistical Theory and Practice
(2020) 14:60
There are however experimental situations where specific experimental constraints (cost, time, infeasible experimentation at specific level combinations of factors) raise the need for using a design matrix which is not the full (and possibly replicated) factorial design. In such a design matrix, it is possible for a specific combination of the levels of the factors to appear as a run more times than the others or not at all. Such a collection of runs is generally called a fractional factorial design. Fractional factorial designs are the most popular choices for experimentation in various fields, mainly due to their flexibility in run size. The use of any randomly selected collection of runs out of the different combinations of the levels of the factors (i.e., an arbitrary formed fractional factorial design) does not usually guarantee an independent estimation of the factorial effects of interest, since a phenomenon that is called aliasing of effects, often occurs. In most experi
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