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Generalized resolution and minimum aberration criteria for plackett-burman and other nonregular factorial designs

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.

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Citations
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Journal ArticleDOI
TL;DR: Important developments in optimality criteria and comparison are reviewed, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.
Abstract: Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main e�ffects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.

73 citations


Cites background or methods from "Generalized resolution and minimum ..."

  • ...To define these two important concepts, generalized resolution and generalized minimum aberration, Deng and Tang (1999) and Tang and Deng (1999) introduced the important notion of J-characteristics and is defined as follows....

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  • ...Deng and Tang (1999) and Tang and Deng (1999) introduced the concepts of generalized resolution and generalized minimum aberration for two-level nonregular designs....

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  • ...If δ > 0, a subset s of D with r columns contains at least Nδ/2r copies of a full 2r factorial and therefore the projectivity of D is at least r (Deng and Tang (1999))....

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  • ...Two regular designs of the same resolution can be distinguished using the minimum aberration criterion, and the same idea can be applied to nonregular designs using the minimum G-aberration criterion (Deng and Tang (1999))....

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  • ...Deng and Tang (1999) provided a statistical justification for the generalized resolution by showing that designs with maximum generalized resolution minimize the contamination of nonnegligible two-factor interactions on the estimation of main effects....

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01 Jan 2002
TL;DR: Tang and Deng as discussed by the authors investigated the relationship between GMA and model-dependent efficiency criteria, and showed that GMA is supported by these criteria, which can be used to classify and rank-order these designs.
Abstract: Tang and Deng (1999) proposed a generalized minimum aberration cri- terion (GMA) as a natural extension of the minimum aberration criterion (MA) from regular to nonregular designs. While MA was defined for regular designs only, GMA applies to both regular and nonregular designs. In this paper, we investigate the relationship between GMA and some model-dependent efficiency criteria, and show that GMA is supported by these criteria. An extensive evaluation of designs with 20 runs and 5 factors shows that the GMA criterion can be used to classify and rank-order these designs. Empirical studies also demonstrate that the GMA ranking is consistent with those obtained by using other model-dependent efficiency criteria.

71 citations


Cites background from "Generalized resolution and minimum ..."

  • ...S)-criterion (Eccleston and Hedayat (1974)), which has long been advocated in the optimal design literature as a good surrogate for such traditional optimality criteria as the D- and A-criteria....

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  • ...Under the hierarchical assumption that lowerorder effects are more important than higher-order effects and that effects of the same order are equally important, a minimum aberration (MA) design is preferred (Fries and Hunter (1980))....

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  • ...Deng and Tang (1999) proposed a generalized minimum aberration criterion, a natural extension of MA from regular to nonregular designs....

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Journal ArticleDOI
TL;DR: In this paper, the goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by Xu and Wu (2001) and general methods for constructing optimal multilevel supersaturated design are proposed.
Abstract: Author(s): Xu, Hongquan; Wu, CFJ | Abstract: A supersaturated design is a design whose run size is not large enough for estimating all the main effects. The goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by Xu and Wu (2001). Optimal supersaturated designs are shown to have a periodic property and general methods for constructing optimal multilevel supersaturated designs are proposed. Inspired by the Addelman-Kempthorne construction of orthogonal arrays, optimal multi-level supersaturated designs are given in an explicit form: columns are labeled with linear or quadratic polynomials and rows are points over a finite field. Additive characters are used to study the properties of resulting designs. Some small optimal supersaturated designs of 3, 4 and 5 levels are listed with their properties.

65 citations

Journal ArticleDOI
TL;DR: The optimal blocking criteria for nonregular two-level designs is extended by adapting the G- and G2-minimum aberration criteria discussed by Tang and Deng, and the notion of “word” is extended to nonregular designs through a polynomial representation of factorial designs.
Abstract: This article discusses the optimal blocking criteria for nonregular two-level designs. We extend the optimal blocking criteria of Cheng and Wu to nonregular designs by adapting the G- and G2-minimum aberration criteria discussed by Tang and Deng. To define word-length pattern for nonregular designs, we extend the notion of “word” to nonregular designs through a polynomial representation of factorial designs. We define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding. We propose four new criteria, which we use to search for optimal blocking schemes of 12-run, 16-run, and 20-run two-level orthogonal arrays.

60 citations

Posted Content
TL;DR: In this paper, a review of recent developments in optimality criteria and comparison of non-regular fractional factorial designs is presented, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies.
Abstract: Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main effects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.

58 citations

References
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Journal ArticleDOI
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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Journal ArticleDOI

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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Journal ArticleDOI
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

Journal ArticleDOI
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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