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

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

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.

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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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Journal Article•DOI•

An effective step-down algorithm for the construction and the identification of nonisomorphic orthogonal arrays

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P. Angelopoulos¹, H. Evangelaras¹, Christos Koukouvinos¹, E. Lappas¹•Institutions (1)

National Technical University of Athens¹

27 Jun 2007-Metrika

TL;DR: In this article, an effective algorithm for the construction and the identification of two-level nonisomorphic orthogonal arrays is presented, and a full catalogue of OO arrays with parameters OA(24,7,2,t), OO(28,6, 2,t) and OA (32,6-2,2.t), t ≥ 2.

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Abstract: In this paper we present an effective algorithm for the construction and the identification of two-level nonisomorphic orthogonal arrays. Using this algorithm, we identify and list a full catalogue of nonisomorphic orthogonal arrays with parameters OA(24,7,2,t), OA(28,6,2,t) and OA(32,6,2,t), t ≥ 2. Some statistical properties of these designs are also considered.

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18 citations

Additional excerpts

...A wide class of such orthogonal arrays with s = 2 levels can be obtained by selecting columns from Hadamard matrices (see Cheng et al. 2002; Deng and Tang 2002; Evangelaras et al. 2004)....
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Journal Article•DOI•

Choosing columns from the 12-run Plackett-Burman design

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Arden Miller¹, Randy R. Sitter²•Institutions (2)

University of Auckland¹, Simon Fraser University²

01 Apr 2004-Statistics & Probability Letters

TL;DR: In this article, the authors focus on the 12-run Plackett-Burman design (PB12) which can be used to estimate the number of 2-factor interactions.

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18 citations

Journal Article•DOI•

Two-Level Orthogonal Screening Designs With 24, 28, 32, and 36 Runs

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Eric D. Schoen¹, Nha Vo-Thanh¹, Peter Goos¹•Institutions (1)

University of Antwerp¹

20 Jan 2017-Journal of the American Statistical Association

TL;DR: In this article, the potential of two-level orthogonal designs to fit models with main effects and two-factor interaction effects is commonly assessed through the correlation between contrast vectors involving these effects.

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Abstract: The potential of two-level orthogonal designs to fit models with main effects and two-factor interaction effects is commonly assessed through the correlation between contrast vectors involving these effects. We study the complete catalog of nonisomorphic orthogonal two-level 24-run designs involving 3–23 factors and we identify the best few designs in terms of these correlations. By modifying an existing enumeration algorithm, we identify the best few 28-run designs involving 3–14 factors and the best few 36-run designs in 3–18 factors as well. Based on a complete catalog of 7570 designs with 28 runs and 27 factors, we also seek good 28-run designs with more than 14 factors. Finally, starting from a unique 31-factor design in 32 runs that minimizes the maximum correlation among the contrast vectors for main effects and two-factor interactions, we obtain 32-run designs that have low values for this correlation. To demonstrate the added value of our work, we provide a detailed comparison of our desi...

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18 citations

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

...To evaluate designs, we consider their strength (Hedayat, Sloane, and Stufken 1999), their confounding frequency vector (CFV; Deng and Tang 1999, 2002), and their generalized word-length pattern (GWLP; Tang and Deng 1999)....
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...The article therefore reports designs that perform well in terms of the G-aberration (Deng and Tang 1999, 2002) and G2-aberration (Tang and Deng 1999) criteria....
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...Deng and Tang (1999) showed that the J3-characteristics in two-level strength-2 designs are of the form N − 8k, where k is a nonnegative integer, N is the run size, and k ≤ N/8....
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Journal Article•DOI•

Generalized resolution for orthogonal arrays

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Ulrike Grömping, Hongquan Xu

28 May 2014-arXiv: Statistics Theory

TL;DR: In this paper, the authors derived two versions of generalized resolution for qualitative factors, both of which are generalizations of the generalized resolution by Deng and Tang [Statist. Sinica 9 (1999) 1071-1082] and Tang and Deng [Ann. Statist. 27 (1999] 1914-1926].

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Abstract: The generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion (Xu and Wu [Ann. Statist. 29 (2001) 1066-1077]). We provide a statistical interpretation for the number of shortest words of an orthogonal array in terms of sums of $R^2$ values (based on orthogonal coding) or sums of squared canonical correlations (based on arbitrary coding). Directly related to these results, we derive two versions of generalized resolution for qualitative factors, both of which are generalizations of the generalized resolution by Deng and Tang [Statist. Sinica 9 (1999) 1071-1082] and Tang and Deng [Ann. Statist. 27 (1999) 1914-1926]. We provide a sufficient condition for one of these to attain its upper bound, and we provide explicit upper bounds for two classes of symmetric designs. Factor-wise generalized resolution values provide useful additional detail.

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17 citations

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

...Section 3 briefly introduces generalized resolution by Deng and Tang (1999) and Tang and Deng (1999) and generalizes it in two meaningful ways....
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...We have provided a statistically meaningful interpretation for the building blocks of GWLP and have generalized resolution by Deng and Tang (1999) and Tang and Deng (1999) in two meaningful ways for qualitatitve factors....
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...For 2-level designs, their approach boils down to omitting the square root from √ max(u1,...,uR) aR(u1, . . . , uR) in (4), which implies that their proposal does not simplify to the well-grounded generalized resolution of Deng and Tang (1999)/Tang and Deng (1999) for 2-level designs....
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...Deng and Tang (1999) looked at the absolute sums of the columns of M, which were termed J -characteristics by Tang and Deng (1999)....
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...Before presenting the new proposals for generalized resolution, we briefly review generalized resolution for symmetric 2-level designs by Deng and Tang (1999) and Tang and Deng (1999)....
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Journal Article•DOI•

De-aliasing effects using semifoldover techniques

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Narayanaswamy Balakrishnan¹, Po Yang²•Institutions (2)

McMaster University¹, DePaul University²

01 Sep 2009-Journal of Statistical Planning and Inference

TL;DR: In this article, the authors investigate various semifoldover designs obtained from a general two-level factorial design, and discuss when a main factor or a two-factor interaction can be de-aliased from their aliased two factor interactions.

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17 citations

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References

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Open Access

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Journal Article•DOI•

The design of optimum multifactorial experiments

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R. L. Plackett, J. P. Burman

01 Jun 1946-Biometrika

3,546 citations

Journal Article•DOI•

A Basis for the Selection of a Response Surface Design

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George E. P. Box¹, Norman R. Draper²•Institutions (2)

Princeton University¹, University of North Carolina at Chapel Hill²

01 Sep 1959-Journal of the American Statistical Association

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.

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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...

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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 Article•DOI•

The 2 k — p Fractional Factorial Designs

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George E. P. Box¹, J. S. Hunter¹•Institutions (1)

University of Wisconsin-Madison¹

01 Aug 1961-Technometrics

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 Article•DOI•

The 2 k-p fractional factorial designs part I

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George E. P. Box¹, J. S. Hunter¹•Institutions (1)

University of Wisconsin-Madison¹

01 Feb 2000-Technometrics

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.

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Abstract: (2000). The 2 k—p Fractional Factorial Designs Part I. Technometrics: Vol. 42, No. 1, pp. 28-47.

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449 citations

Journal Article•DOI•

Minimum Aberration 2 k–p Designs

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Arthur Fries, William G. Hunter

01 Nov 1980-Technometrics

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.

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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.

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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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