A Modern Theory of Factorial Designs by Rahul Mukerjee, C.F. J. Wu

By Rahul Mukerjee, C.F. J. Wu

The final 20 years have witnessed an important progress of curiosity in optimum factorial designs, lower than attainable version uncertainty, through the minimal aberration and similar standards. This ebook supplies, for the 1st time in ebook shape, a finished and up to date account of this contemporary concept. Many significant sessions of designs are coated within the ebook. whereas retaining a excessive point of mathematical rigor, it additionally presents wide layout tables for examine and useful reasons. except being invaluable to researchers and practitioners, the publication can shape the middle of a graduate point path in experimental layout.

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3) is only necessary but not sufficient for the existence of a design of resolution four or higher. 2 has another important implication from traditional optimality considerations. To motivate the ideas, consider an sn−k design d(B) of resolution three. 1, d(B) allows estimation of all main effect contrasts under the absence of all interactions. One can, however, consider any other selection of sn−k treatment combinations and wonder about the performance of d(B), vis-` a-vis any such rival fraction, for the estimation of the main effect contrasts.

Consider the pencil b = (1, 0, 0, 0, 0) , which is not a defining pencil. 11) where λ1 , λ2 ∈ {0, 1}. Considering all possible choices of λ1 and λ2 , this alias set turns out to be {(1, 0, 0, 0, 0) , (0, 1, 0, 1, 0) , (0, 0, 1, 0, 1) , (1, 1, 1, 1, 1) }. 7), the above alias set can as well be described as 1 = 24 = 35 = 12345. 7) by multiplying the latter by 1, which stands for the pencil (1, 0, 0, 0, 0) . This multiplication must follow the convention that (i) any squared symbol is dropped, and (ii) any string of symbols is invariant under multiplication by I.

Two such conditions are presented in the next result. 3. Let d(B) be an sn−k design of resolution R. (a) For R ≥ 3, sn−k − 1 . 2) sn−k−1 − 1 + 1. 3) n≤ (b) For R ≥ 4, n≤ Proof. 2, the treatment combinations in d(B) form an OA(sn−k , n, s, 2). 1, the result follows. 1 (b), sn−k ≥ 1 + n(s − 1) + (n − 1)(s − 1)2 , which, on simplification, yields the desired inequality. 2) is also sufficient for the existence of a design of resolution three or higher. 3) for s = 2 will also be presented there. 3) is not sufficient for the existence of a design of resolution four or higher.

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