Generalized resolution and minimum aberration criteria for plackett-burman and other nonregular factorial designs
01 Jan 1999-
TL;DR: In this paper, a generalized resolution criterion is defined and used for assessing non-regular fractional factorials, notably Plackett-Burman designs, which is intended to capture projection properties, complementing that of Webb (1964) whose concept of resolution concerns the estimability of lower order fractional fractional factors under the assumption that higher order effects are negligible.
Abstract: Resolution has been the most widely used criterion for comparing regular fractional factorials since it was introduced in 1961 by Box and Hunter. In this pa- per, we examine how a generalized resolution criterion can be defined and used for assessing nonregular fractional factorials, notably Plackett-Burman designs. Our generalization is intended to capture projection properties, complementing that of Webb (1964) whose concept of resolution concerns the estimability of lower order ef- fects under the assumption that higher order effects are negligible. Our generalized resolution provides a fruitful criterion for ranking different designs while Webb's resolution is mainly useful as a classification rule. An additional advantage of our approach is that the idea leads to a natural generalization of minimum aberration. Examples are given to illustrate the usefulness of the new criteria.
Citations
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TL;DR: In this paper, a number of methods have been proposed to identify and estimate active interaction effects via non-regular factorial designs using Plackett-Burman designs, and the purpose of this article is to suggest approaches that the author has found useful.
Abstract: Methods have recently been proposed to identify and estimate active interaction effects via nonregular factorial designs using Plackett-Burman designs. The purpose of this article is to suggest approaches that the author has found useful.
13 citations
Cites background from "Generalized resolution and minimum ..."
...Deng and Tang (1999) and Tang and Deng (1999) initiated this fundamental breakthrough, introducing generalized resolution, minimum G-aberration and minimum G2aberration criteria....
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TL;DR: In this article, it was shown that for a given even number of rows, there is just one isomorphism class for conference designs with two or three columns, and a classification criterion for definitive screening designs founded on projections into four factors.
Abstract: A conference design is a rectangular matrix with orthogonal columns, one zero in each column, at most one zero in each row and -1's and +1's elsewhere. A definitive screening design can be constructed by folding over a conference design and adding a row vector of zeroes. We prove that, for a given even number of rows, there is just one isomorphism class for conference designs with two or three columns. Next, we derive all isomorphism classes for conference designs with four columns. Based on our results, we propose a classification criterion for definitive screening designs founded on projections into four factors. We illustrate the potential of the criterion by studying designs with 24 and 82 factors.
13 citations
Cites background from "Generalized resolution and minimum ..."
...Deng and Tang (1999) also point out that the J3-characteristic of any z-row orthogonal array involving three two-level factors is a multiple of 4, imsart-aos ver....
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...Deng and Tang (1999) also point out that the J3-characteristic of any z-row orthogonal array involving three two-level factors is a multiple of 4,...
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...The generalized aberration criterion for orthogonal two-level designs of Deng and Tang (2002) is based on the confounding frequency vector (F3, F4, ....
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...Accordingly, projection properties form the basis of the generalized aberration criterion for classifying orthogonal arrays (Deng and Tang, 1999)....
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TL;DR: In this paper, the authors focus on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal quaternary codes have larger generalized resolution and projectivity than comparable regular designs.
Abstract: The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations. We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs. Moreover, some of these designs are found to have maximum projectivity among all designs.
12 citations
Cites background from "Generalized resolution and minimum ..."
...The notions of resolution and aberration have been generalized, with statistical justifications, to these designs; see [7, 9, 13, 19, 20, 27, 28]....
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...The generalized resolution [9] of D is defined as R(D) = r+1− max #S=r ρr(S;D)....
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...As shown in [9], a design with resolution R > r has projectivity greater than r....
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Posted Content•
TL;DR: In this paper, the problem of finding all non-isomorphic solutions of two novel integer linear programming formulations for classifying all nonisomorphic OA(N,k,s,t) given a set of all OAs (N, k-1, s,t).
Abstract: Classifying orthogonal arrays is a well known important class of problems that asks for finding all non-isomorphic, non-negative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods based on finding all non-isomorphic solutions of two novel integer linear programming formulations for classifying all non-isomorphic OA(N,k,s,t) given a set of all non-isomorphic OA(N,k-1,s,t). We also establish the concept of orthogonal design equivalence of OA(N,k,2,t) to reduce the number of integer linear programs (ILPs) whose all non-isomorphic solutions need to be enumerated by our methods. For each ILP, we determine the largest group of permutations that can be exploited with the branch-and-bound (B&B) with isomorphism pruning algorithm of Margot [Discrete Optim~4 (2007), 40-62] without losing isomorphism classes of OA(N,k,2,t). Our contributions brought the classifications of all non-isomorphic OA(160,k,2,4) for k=9,10 and OA(176,k,2,4) for k=5,6,7,8,9,10 within computational reach. These are the smallest s=2, t=4 cases for which classification results are not available in the literature.
12 citations
TL;DR: In this paper, a kind of indicator function based on orthogonal complex contrasts is introduced to represent general factorial designs and its significance on projection designs is presented, and a generalized resolution and a new aberration criterion are developed to rank combinatorially non-isomorphic designs with prime levels.
12 citations
Cites background from "Generalized resolution and minimum ..."
...As discussed in Deng and Tang (1999) , the resolution of a regular fractional factorial design is meaningful in two ways: projection viewpoint and estimability viewpoint....
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...The resolution of nonregular designs got a breakthrough in Deng and Tang (1999) ....
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References
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3,546 citations
TL;DR: In this paper, the problem of choosing a design such that the polynomial f(ξ) = f (ξ1, ξ2, · · ·, ξ k ) fitted by the method of least squares most closely represents the true function over some region of interest R in the ξ space, no restrictions being introduced that the experimental points should necessarily lie inside R, is considered.
Abstract: The general problem is considered of choosing a design such that (a) the polynomial f(ξ) = f(ξ1, ξ2, · · ·, ξ k ) in the k continuous variables ξ' = (ξ1, ξ2, · · ·, ξ k ) fitted by the method of least squares most closely represents the true function g(ξ1, ξ2, · · ·, ξ k ) over some “region of interest” R in the ξ space, no restrictions being introduced that the experimental points should necessarily lie inside R; and (b) subject to satisfaction of (a), there is a high chance that inadequacy of f(ξ) to represent g(ξ) will be detected. When the observations are subject to error, discrepancies between the fitted polynomial and the true function occur: i. due to sampling error (called here “variance error”), and ii. due to the inadequacy of the polynomial f(ξ) exactly to represent g(ξ) (called here “bias error”). To meet requirement (a) the design is selected so as to minimize J, the expected mean square error averaged over the region R. J contains two components, one associated entirely with varian...
697 citations
"Generalized resolution and minimum ..." refers result in this paper
...Finally, we note that our argument for minimizing biases is similar to that in Box and Draper (1959)....
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471 citations
"Generalized resolution and minimum ..." refers methods in this paper
...For a detailed discussion on the concept of resolution for regular factorials, we refer to Box and Hunter (1961)....
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TL;DR: The 2 k-p Fractional Factorial Designs Part I. as discussed by the authors is a collection of fractional fractional factorial designs with a focus on the construction of the construction.
Abstract: (2000). The 2 k—p Fractional Factorial Designs Part I. Technometrics: Vol. 42, No. 1, pp. 28-47.
449 citations
TL;DR: In this article, the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution, and algorithms are presented for constructing these minimum aberration designs.
Abstract: For studying k variables in N runs, all 2 k–p designs of maximum resolution are not equally good. In this paper the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution. Algorithms are presented for constructing these minimum aberration designs.
420 citations
"Generalized resolution and minimum ..." refers methods in this paper
...For results on minimum aberration designs, we refer to Fries and Hunter (1980), Franklin (1984), Chen and Wu (1991), Chen (1992), Tang and Wu (1996), Chen and Hedayat (1996) and Cheng, Steinberg and Sun (1999)....
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