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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Minimum moment aberration for nonregular designs and supersaturated designs

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Hongquan Xu¹•Institutions (1)

University of California, Los Angeles¹

01 Jan 2001

TL;DR: In this paper, a new combinatorial criterion, called minimum moment aberration, is proposed for assessing the goodness of nonregular designs and supersaturated designs, which is a good surrogate with tremendous computational advantages for many statistically justified criteria, such as minimum G2-aberrration, generalized minimum aberration and E(s2).

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Abstract: Nonregular designs are used widely in experiments due to their run size economy and flexibility. These designs include the Plackett-Burman designs and many other symmetrical and asymmetrical orthogonal arrays. Supersaturated designs have become increasingly popular in recent years because of the potential in saving run size and its technical novelty. In this paper, a novel combinatorial criterion, called minimum moment aberration, is proposed for assessing the goodness of nonregular designs and supersaturated designs. The new criterion, which is to sequentially minimize the power moments of the number of coincidence among runs, is a good surrogate with tremendous computational advantages for many statistically justified criteria, such as minimum G2-aberrration, generalized minimum aberration and E(s2). In addition, the minimum moment aberration is conceptually simple and convenient for theoretical development. The general theory developed here not only unifies several separate results, but also provides many novel results on nonregular designs and supersaturated designs.

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152 citations

Journal Article•DOI•

Uniform designs limit aliasing

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Fred J. Hickernell, Min-Qian Liu

01 Dec 2002-Biometrika

TL;DR: In this article, it is shown that uniform designs limit the effects of aliasing to yield reasonable efficiency and robustness together, while robust experimental designs guard against inaccurate estimates caused by model misspecification.

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Abstract: SUMMARY When fitting a linear regression model to data, aliasing can adversely affect the estimates of the model coefficients and the decision of whether or not a term is significant. Optimal experimental designs give efficient estimators assuming that the true form of the model is known, while robust experimental designs guard against inaccurate estimates caused by model misspecification. Although it is rare for a single design to be both maximally efficient and robust, it is shown here that uniform designs limit the effects of aliasing to yield reasonable efficiency and robustness together. Aberration and resolution measure how well fractional factorial designs guard against the effects of aliasing. Here it is shown that the definitions of aberration and resolution may be generalised to other types of design using the discrepancy.

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126 citations

Journal Article•DOI•

An Algorithm for Constructing Orthogonal and Nearly Orthogonal Arrays with Mixed Levels and Small Runs

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Hongquan Xu¹•Institutions (1)

University of California, Los Angeles¹

01 Nov 2002-Technometrics

TL;DR: This article describes a simple and effective algorithm for constructing mixed-level orthogonal and nearly-orthogonal arrays that can construct a variety of small-run designs with good statistical properties efficiently.

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Abstract: Orthogonal arrays are used widely in manufacturing and high-technology industries for quality and productivity improvement experiments. For reasons of run size economy or flexibility, nearly-orthogonal arrays are also used. The construction of orthogonal or nearly-orthogonal arrays can be quite challenging. Most existing methods are complex and produce limited types of arrays. This article describes a simple and effective algorithm for constructing mixed-level orthogonal and nearly-orthogonal arrays that can construct a variety of small-run designs with good statistical properties efficiently.

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122 citations

Additional excerpts

...(See Lin and Draper 1992, Wang and Wu 1995, Cheng 1995, Box and Tyssedal 1996, Deng and Tang 1999, Tang and Deng 1999, and Xu and Wu 2001 for classi cation or discrimination of OAs.)...
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Journal Article•DOI•

A quality by design (QbD) case study on liposomes containing hydrophilic API: II. Screening of critical variables, and establishment of design space at laboratory scale.

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Xiaoming Xu¹, Mansoor A. Khan², Diane J. Burgess¹•Institutions (2)

University of Connecticut¹, Silver Spring Networks²

28 Feb 2012-International Journal of Pharmaceutics

TL;DR: Two statistical designs were used in this case study as part of an investigation into the feasibility and the advantages of applying QbD concepts to liposome-based complex parenteral controlled release systems containing a hydrophilic active pharmaceutical ingredient (API).

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109 citations

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

...Additional data points (white disks) are include eferences to color in this figure legend, the reader is referred to the web version of Deng and Tang, 1999) with very high efficiency and accuracy, but t cannot separate the main effects from the possible interactions. owever, as the goal of this…...
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Journal Article•DOI•

Indicator function and its application in two-level factorial designs

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Kenny Q. Ye¹•Institutions (1)

Stony Brook University¹

01 Jun 2003-Annals of Statistics

TL;DR: In this paper, a polynomial indicator function is used to generalize the aberration criterion of a regular two-level fractional factorial design to all 2-level factorial designs and an important identity of generalized aberration is proved.

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Abstract: A two-level factorial design can be uniquely represented by a polynomial indicator function. Therefore, properties of factorial designs can be studied through their indicator functions. This paper shows that the indicator function is an effective tool in studying two-level factorial designs. The indicator function is used to generalize the aberration criterion of a regular two-level fractional factorial design to all two-level factorial designs. An important identity of generalized aberration is proved. The connection between a uniformity measure and aberration is also extended to all two-level factorial designs.

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105 citations

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

...What should be mentioned here are the J−Characteristics used by Deng and Tang (1999) as building blocks in defining their generalized aberration criterion....
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...(Deng and Tang, 1999) Regard a n × s design as a set of s columns A = {c1, c2, · · · , cs}....
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...Recently, Deng and Tang (1999) and Tang and Deng (2000) generalize resolution and aberration criterion to nonregular two-level designs based on the J−Characteristics....
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References

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Journal Article•DOI•

Projective properties of certain orthogonal arrays

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George E. P. Box¹, John Tyssedal•Institutions (1)

University of Wisconsin-Madison¹

01 Dec 1996-Biometrika

TL;DR: In this article, it was shown that doubling an n x n orthogonal array is always of projectivity P = 2, and that a two-level cyclic design is either a factorial array, and hence has P = 3, or a 2-cycle cyclic with 4m runs, m odd, has p = 3.

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Abstract: SUMMARY A question of importance in factor screening is when a two-level orthogonal design for a multifactor experiment can be projected into lower dimension, typically P = 2 or 3. New results relate to the projectivity P of saturated designs in which n - 1 factors are tested in n runs. It is shown that: a design obtained by 'doubling' an n x n orthogonal array is always of projectivity P = 2; a twolevel cyclic design is either a factorial array, and hence has P = 2, or it has P = 3; a two-level orthogonal design with 4m runs, m odd, has P = 3. In particular these results allow the designs derived by Plackett & Burman (1946) to be categorised in terms of these projective properties.

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113 citations

"Generalized resolution and minimum ..." refers background in this paper

...Thus the projectivity, as defined in Box and Tyssedal (1996), of the design is r....
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Journal Article•DOI•

Characterization of minimum aberration $2\sp {n-k}$ designs in terms of their complementary designs

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Boxin Tang, C. F. J. Wu

01 Dec 1996-Annals of Statistics

TL;DR: In this paper, a general result is obtained that relates the word-length pattern of a design to that of its complementary design, which is used to characterize minimum aberration designs in terms of properties of their complementary designs.

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Abstract: A general result is obtained that relates the word-length pattern of a $2^{n-k}$ design to that of its complementary design. By applying this result and using group isomorphism, we are able to characterize minimum aberration $2^{n-k}$ designs in terms of properties of their complementary designs. The approach is quite powerful for small values of $2^{n-k} - n - 1$. In particular, we obtain minimum aberration $2^{n-k}$ designs with $2^{n-k} - n - 1 = 1$ to 11 for any n and k.

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111 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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Journal Article•DOI•

Some Projection Properties of Orthogonal Arrays

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Ching-Shui Cheng

01 Aug 1995-Annals of Statistics

TL;DR: In this paper, it was shown that the projection of an orthogonal array with two-level factors onto any t + 1 factor can be one of three types: one or more copies of the complete $2^{t+1} factorial, a half-replicate of the factorial or a combination of both.

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Abstract: The definition of an orthogonal array imposes an important geometric property: the projection of an $\mathrm{OA}(\lambda 2^t, 2^k, t)$, a $\lambda 2^t$-run orthogonal array with $k$ two-level factors and strength $t$, onto any $t$ factors consists of $\lambda$ copies of the complete $2^t$ factorial. In this article, projections of an $\mathrm{OA}(N, 2^k, t)$ onto $t + 1$ and $t + 2$ factors are considered. The projection onto any $t + 1$ factors must be one of three types: one or more copies of the complete $2^{t+1}$ factorial, one or more copies of a half-replicate of $2^{t+1}$ or a combination of both. It is also shown that for $k \geq t + 2$, only when $N$ is a multiple of $2^{t+1}$ can the projection onto some $t + 1$ factors be copies of a half-replicate of $2^{t+1}$. Therefore, if $N$ is not a multiple of $2^{t+1}$, then the projection of an $\mathrm{OA}(N, 2^k, t)$ with $k \geq t + 2$ onto any $t + 1$ factors must contain at least one complete $2^{t+1}$ factorial. Some properties of projections onto $t + 2$ factors are established and are applied to show that if $N$ is not a multiple of 8, then for any $\mathrm{OA}(N, 2^k, 2)$ with $k \geq 4$, the projection onto any four factors has the property that all the main effects and two-factor interactions of these four factors are estimable when the higher-order interactions are negligible.

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107 citations

"Generalized resolution and minimum ..." refers background or methods or result in this paper

...2 of Cheng (1995), there does not exist an s with three columns satisfying J3(s) = n because n is not a multiple of 8....
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...The results in Proposition 2 are essentially in Cheng (1995). We note that part (ii) of Proposition 2 is more explicit than that given by Cheng (1995). In the discussion following his Theorem 2.1, Cheng (1995) stated that, in our notation, s contains copies of a complete 2r factorial plus copies of a half replicate and did not give the explicit numbers of copies in both cases....
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...The results in Proposition 2 are essentially in Cheng (1995). We note that part (ii) of Proposition 2 is more explicit than that given by Cheng (1995). In the discussion following his Theorem 2....
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...Cheng (1995) provided some general results on the projection properties of nonregular factorials, and these results cover as special cases some of the computer findings given in Lin and Draper (1992) and in Wang and Wu (1995)....
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...Cheng (1995) provided some general results on the projection properties of nonregular factorials, and these results cover as special cases some of the computer findings given in Lin and Draper (1992) and in Wang and Wu (1995). A basic problem in this area remains unsolved, or at least has not been systematically attempted, despite the above important contributions....
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Journal Article•DOI•

Constructing Tables of Minimum Aberration pn-m Designs

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M. F. Franklin

01 Aug 1984-Technometrics

TL;DR: In this paper, the Fries and Hunter algorithm was used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime, and a matrix is given for generating 3 n-m designs with m, n ≤ 6, which have, or nearly have, minimum aberration.

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Abstract: Fries and Hunter (1980) presented a practical algorithm for selecting standard 2 n–m fractional factorial designs based on a criterion they called “minimum aberration.” In this article some simple results are presented that enable the Fries–Hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime. Examples of minimum aberration 2 n–m designs with resolution R ≥ 4 are given for m, n – m < 9. A matrix is given for generating 3 n–m designs with m, n – m ≤ 6, which have, or nearly have, minimum aberration.

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101 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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Journal Article•DOI•

2n-l designs with weak minimum aberration

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Hegang Chen¹, A. S. Hedayat•Institutions (1)

Case Western Reserve University¹

01 Dec 1996-Annals of Statistics

TL;DR: In this paper, the concept of weak minimum aberration was introduced and several families of fractional factorial designs of resolution III and IV with minimal aberration were obtained. But they did not show the usefulness of this new design concept.

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Abstract: Since not all $2^{n-1}$ fractional factorial designs with maximum resolution are equally good, Fries and Hunter introduced the minimum aberration criterion for selecting good $2^{n-1}$ fractional factorial designs with the same resolution. We modify the concept of minimum aberration and define weak minimum aberration and show the usefulness of this new design concept. Using some techniques from finite geometry, we construct $2^{n-1}$ fractional factorial designs of resolution III with weak minimum aberration. Further, several families of $2^{n-1}$ fractional factorial designs of resolution III and IV with minimum aberration are obtained.

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89 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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