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.
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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).
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.
152 citations
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.
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.
126 citations
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.
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.
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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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).
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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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.
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.
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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TL;DR: This paper presents a large number of designs of Plackett-Burman designs that have been used in screening experiments for identifying important main effects and some of them have been criticized for their complex aliasing patterns.
Abstract: Traditionally, Plackett-Burman (PB) designs have been used in screening experiments for identifying important main effects. The PB designs whose run sizes are not a power of two have been criticized for their complex aliasing patterns, which according t..
354 citations
"Generalized resolution and minimum ..." refers background in this paper
...Hamada and Wu (1992) showed that for data from designs with complex aliasing, it is possible to detect interaction effects....
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01 Jan 1980
TL;DR: The concept of resolution was introduced by Box and Hunter as discussed by the authors, who defined the resolution of a two-level fractional factorial design as the length of the shortest word in the defining relation.
Abstract: Fractional factorial designs-especially the twolevel designs-are useful in a variety of experimental situations, for example, (i) screening studies in which only a subset of the variables is expected to be important, (ii) research investigations in which certain interactions are expected to be negligible and (iii) experimental programs in which groups of runs are to be performed sequentially, ambiguities being resolved as the investigation evolves (see Box, Hunter and Hunter, 1978). The literature on fractional factorial designs is extensive. For references before 1969, see the comprehensive bibliography of Herzberg and Cox (1969). For more recent references, see Daniel (1976) and Joiner (1975-79). A useful concept associated with 2k-P fractional factorial designs is that of resolution (Box and Hunter, 1961). A design is of resolution R if no cfactor effect is confounded with any other effect containing less than R c factors. For example, a design of resolution III does not confound main effects with one another but does confound main effects with two-factor interactions, and a design of resolution IV does not confound main effects with two-factor interactions but does confound two-factor interactions with one another. The resolution of a two-level fractional factorial design is the length of the shortest word in the defining relation. Usually an experimenter will prefer to use a design which has the highest
354 citations
"Generalized resolution and minimum ..." refers background or 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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...Hamada and Wu (1992) showed that for data from designs with complex aliasing, it is possible to detect interaction effects....
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...Hamada and Wu (1992) showed that for data from designs with complex aliasing, it is possible to detect interaction effects. Lin and Draper (1992) studied the projection properties of some Plackett-Burman designs, and this line of research was further pursued and explored from a different angle by Wang and Wu (1995) under the term “hidden projection”....
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TL;DR: In this article, the maximum number of factors that can be accommodated when the degree of the nonorthogonality is specified is examined, and interesting properties of systematic supersaturated designs are revealed.
Abstract: Practitioners are routinely faced with distinguishing between factors that have real effects and those whose apparent effects are due to random error. When there are many factors, the usual advice given is to run so-called main-effect designs (Resolution III designs in the orthogonal case), that require at least k + 1 runs for investigating k factors. This may be wasteful, however, if the goal is only to detect those active factors. This is particularly true when the number of factors is large. In such situations, a supersaturated design can often save considerable cost. A supersaturated design is a (fraction of a factorial) design composed of n observations where n < k + 1. When such a design is used, the abandonment of orthogonality is inevitable. This article examines the maximum number of factors that can be accommodated when the degree of the nonorthogonality is specified. Furthermore, interesting properties of systematic supersaturated designs are revealed. For example, such a design may be adequate...
198 citations
Additional excerpts
...For example, see Lin (1995)....
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TL;DR: The Plackett and Burman designs also have interesting projective properties, knowledge of which allows the experimenter to follow up an initial PLACKET and BurMAN design with runs that increase the initial resolution for the factors that appear to matter and thus permit efficient separation of effects of interest.
Abstract: The projection properties of the 2 R q–p fractional factorials are well known and have been used effectively in a number of published examples of experimental investigations. The Plackett and Burman designs also have interesting projective properties, knowledge of which allows the experimenter to follow up an initial Plackett and Burman design with runs that increase the initial resolution for the factors that appear to matter and thus permit efficient separation of effects of interest. Projections of designs into 2–5 dimensions are discussed, and the 12-run case is given in detail. A numerical example illustrates the practical uses of these projections.
164 citations
"Generalized resolution and minimum ..." refers background or result in this paper
...In fact, these are the three nonequivalent projection designs onto k = 4 dimensions discovered by Lin and Draper (1992)....
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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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...Lin and Draper (1992) studied the projection properties of some Plackett-Burman designs, and this line of research was further pursued and explored from a different angle by Wang and Wu (1995) under the term “hidden projection”....
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...According to Lin and Draper (1992), these are the only two nonequivalent projection designs onto five factors....
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TL;DR: In this article, the performance of minimum aberration two-level fractional factorial designs is studied under two criteria of model robustness, i.e., the number of aliases of main effects and the sum of squares of the sizes of alias sets of two-factor interactions.
Abstract: Summary. The performance of minimum aberration two-level fractional factorial designs is studied under two criteria of model robustness. Simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived. It is also shown that a minimum aberration design of resolution IlIl or higher maximizes the number of two-factor interactions which are not aliases of main effects and, subject to that condition, minimizes the sum of squares of the sizes of alias sets of two-factor interactions. This roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform. It follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here. Examples are given to show that minimum aberration designs are indeed highly efficient.
139 citations