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•

A fresh look at effect aliasing and interactions: some new wine in old bottles

[...]

C. F. Jeff Wu¹•Institutions (1)

Georgia Institute of Technology¹

09 Feb 2018-Annals of the Institute of Statistical Mathematics

TL;DR: It is argued that the “de-aliasing” of aliased effects can indeed be resolved by employing a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality.

...read moreread less

Abstract: Interactions and effect aliasing are among the fundamental concepts in experimental design. In this paper, some new insights and approaches are provided on these subjects. In the literature, the “de-aliasing” of aliased effects is deemed to be impossible. We argue that this “impossibility” can indeed be resolved by employing a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality. This approach is successfully applied to three classes of designs: regular and nonregular two-level fractional factorial designs, and three-level fractional factorial designs. For reparametrization, the notion of conditional main effects (cme’s) is employed for two-level regular designs, while the linear-quadratic system is used for three-level designs. For nonregular two-level designs, reparametrization is not needed because the partial aliasing of their effects already induces non-orthogonality. The approach can be extended to general observational data by using a new bi-level variable selection technique based on the cme’s. A historical recollection is given on how these ideas were discovered.

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3 citations

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

...earchers started to look for extensions of the minimum aberration criterion to nonregular designs. Among them, two major criteria are the generalized minimum aberration criterion (Tang and Deng 1999; Deng and Tang 1999; Xu and Wu 2001) and the minimum moment criterion (Xu 2003). A survey of these advances can be found in Chapter 10 of Cheng (2014). 4 3k−q designs: design classiﬁcation and analysis In this section I...
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Posted Content•

A canonical form for non-regular arrays based on generalized word length pattern values of delete-one-factor projections

[...]

Pieter T. Eendebak

01 Apr 2014-Research Papers in Economics

TL;DR: It is shown that the delete-one-factor projection GWLP method can be adopted to reduce both regular and non-regular orthogonal arrays to canonical form, and the new canonical form is used in an existing framework to generate minimal complete sets of non-symmetricnon-regular arrays.

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Abstract: We introduce a canonical representative for the isomorphism classes of non-regular orthogonal arrays based on the generalized word length patterns (GWLP) of delete-one-factor projections. These GWLP values have been used recently to introduce a fast isomorphism test for two-level regular arrays. We show that the delete-one-factor projection GWLP method can be adopted to reduce both regular and non-regular orthogonal arrays to canonical form. The new canonical form is used in an existing framework to generate minimal complete sets of non-symmetric non-regular arrays. We show that the new method is efficient for reduction to canonical form, but not suitable for generating minimal complete sets.

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3 citations

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

...The resulting designs are then classified with the extended word-length pattern [3]....
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...Array index Group structure Group size 0 [8, 1] 40320 1 [6, 1, 2] 1440 2 [3, 4, 2] 288 3 [7, 2] 10080 4 [6, 2, 1] 1440 5 [8, 1] 40320 6 [5, 1, 2, 1] 240 7 [2, 4, 2, 1] 96 8 [6, 2, 1] 1440 9 [4, 1, 4] 576 10 [5, 2, 2] 480 11 [5, 2, 2] 480 12 [6, 2, 1] 1440 13 [8, 1] 40320 14 [3, 2, 4] 288 15 [9] 362880 16 [9] 362880 17 [9] 362880 18 [4, 3, 1, 1] 144 19 [8, 1] 40320 20 [4, 1, 4] 576 21 [4, 2, 2, 1] 96 22 [4, 2, 2, 1] 96 23 [3, 4, 1, 1] 144 24 [4, 4, 1] 576 25 [4, 4, 1] 576 26 [8, 1] 40320 27 [3, 2, 4] 288 28 [9] 362880 29 [9] 362880 30 [7, 1, 1] 5040 31 [8, 1] 40320 32 [8, 1] 40320 33 [8, 1] 40320...
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Posted Content•DOI•

Some characterizations of affinely full-dimensional factorial designs

[...]

Satoshi Aoki¹, Akimichi Takemura²•Institutions (2)

Kagoshima University¹, University of Tokyo²

01 Dec 2008-arXiv: Methodology

TL;DR: In this paper, a new class of two-level non-regular fractional factorial designs is defined, called affinely full-dimensional factorial design, and the authors investigate the property of this class from the viewpoint of $D$-optimality.

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Abstract: A new class of two-level non-regular fractional factorial designs is defined. We call this class an {\it affinely full-dimensional factorial design}, meaning that design points in the design of this class are not contained in any affine hyperplane in the vector space over $\mathbb{F}_2$. The property of the indicator function for this class is also clarified. A fractional factorial design in this class has a desirable property that parameters of the main effect model are simultaneously identifiable. We investigate the property of this class from the viewpoint of $D$-optimality. In particular, for the saturated designs, the $D$-optimal design is chosen from this class for the run sizes $r \equiv 5,6,7$ (mod 8).

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3 citations

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

...See [13] and [39] for minimum G2-aberration, [47] for generalized minimum aberration, [46] for minimum moment aberration....
[...]

Posted Content•

A fresh look at effect aliasing and interactions: some new wine in old bottles

[...]

C. F. Jeff Wu¹•Institutions (1)

Georgia Institute of Technology¹

06 Mar 2017-arXiv: Methodology

TL;DR: In this article, a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality is proposed to solve the problem of de-aliasing of aliased effects.

...read moreread less

Abstract: Interactions and effect aliasing are among the fundamental concepts in experimental design. In this paper, some new insights and approaches are provided on these subjects. In the literature, the "de-aliasing" of aliased effects is deemed to be impossible. We argue that this "impossibility" can indeed be resolved by employing a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality. This approach is successfully applied to three classes of designs: regular and nonregular two-level fractional factorial designs, and three-level fractional factorial designs. For reparametrization, the notion of conditional main effects (cme's) is employed for two-level regular designs, while the linear-quadratic system is used for three-level designs. For nonregular two-level designs, reparametrization is not needed because the partial aliasing of their effects already induces non-orthogonality. The approach can be extended to general observational data by using a new bi-level variable selection technique based on the cme's. A historical recollection is given on how these ideas were discovered.

...read moreread less

3 citations

Journal Article•DOI•

Efficient arrangements of two-level orthogonal arrays in two and four blocks

[...]

H. Evangelaras¹, C. Peveretos¹•Institutions (1)

University of Piraeus¹

08 Aug 2017-Statistics

TL;DR: In this paper, the nice properties of two-level orthogonal arrays are taken into consideration and an effective method for arranging experimental runs into two and four blocks of the same size is proposed.

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Abstract: When orthogonal arrays are used in practical applications, it is often difficult to perform all the designed runs of the experiment under homogeneous conditions. The arrangement of factorial runs into blocks is usually an action taken to overcome such obstacles. However, an arbitrary configuration might lead to spurious analysis results. In this work, the nice properties of two-level orthogonal arrays are taken into consideration and an effective method for arranging experimental runs into two and four blocks of the same size is proposed. This method is based on the so-called J-characteristics of the corresponding array. General theoretical results are given for studying up to four experimental factors in two blocks, as well as for studying up to three experimental factors in four blocks. Finally, we provide best blocking arrangements when the number of the factors of interest is larger, by exploiting the known lists of non-isomorphic orthogonal arrays with two levels and various run sizes.

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2 citations

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

...This connection has been pointed out by Tang [15] and Stufken and Tang [18] in justifying that the values of J characteristics uniquely defines a fractional factorial design....
[...]
...It is easy to express all the Jm characteristics, m= 1,2,3 and 4, of the n-run design that appear in the information matrix as linear combinations of the multiplicities ai, i = 1, 2, . . . , 2q and set them equal to a given number (which depends on the number of runs, n, of the array following the result of Deng and Tang [16])....
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...The non-diagonal elements of the information matrix ofmodel (1) are highly related to the J-characteristics of the two-level orthogonal array, as they are defined by Deng and Tang [1] and Tang [15]....
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...Deng and Tang [16] proved interesting results for the values of Jm(S) of an orthogonal array with n runs....
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...This action leads to a system of l ≤ 2q linear-independent equations (it is easy to verify that the coefficients of each equation form a column of a Hadamard matrix of order 2q, see also Stufken and Tang [18] who noted that the vector consisting of all the J characteristics of a design is simply the Hadamard transform of the vector a) whose solutionwill give rise to the unknownmultiplicities ai....
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References

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Journal Article•DOI•

The design of optimum multifactorial experiments

[...]

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

[...]

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

[...]

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)....
[...]

Journal Article•DOI•

The 2 k-p fractional factorial designs part I

[...]

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.

...read moreread less

449 citations

Journal Article•DOI•

Minimum Aberration 2 k–p Designs

[...]

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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