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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: In this article, the authors give an expository review of applications of computational algebraic statistics to design and analysis of fractional factorial experiments based on their recent works, which greatly enlarges the scope of factorial designs.
Abstract: We give an expository review of applications of computational algebraic statistics to design and analysis of fractional factorial experiments based on our recent works. For the purpose of design, the techniques of Grobner bases and indicator functions allow us to treat fractional factorial designs without distinction between regular designs and nonregular designs. For the purpose of analysis of data from fractional factorial designs, the techniques of Markov bases allow us to handle discrete observations. Thus the approach of computational algebraic statistics greatly enlarges the scope of fractional factorial designs.

3 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied all the classes of inequivalent projections of certain two-level orthogonal arrays that arise from Hadamard matrices, using well known statistical criteria, such as generalized resolution and generalized minimum aberration.
Abstract: Suppose a large number of factors (q) is examined in an experimental situation. It is often anticipated that only a few (k) of these will be important. Usually, it is not known which of the q factors will be the important ones, that is, it is not known which k columns of the experimental design will be of further interest. Screening designs are useful for such situations. It is of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties, since it helps experimenters in selecting a screening design with favorable properties. In this paper we study all the classes of inequivalent projections of certain two-level orthogonal arrays that arise from Hadamard matrices, using well known statistical criteria, such as generalized resolution and generalized minimum aberration. We also pay attention to each projection's distinct runs. Results are given for orthogonal arrays with n = 16, 20 and 24 runs, wh...

3 citations

Journal ArticleDOI
TL;DR: In this paper, a maximum estimability (maxest) criterion is proposed for design classification and selection, which is an extension and refinement of Webb's resolution criterion for general factorial designs.

3 citations

Journal ArticleDOI
TL;DR: In this paper, the authors focus on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal quaternary codes often have larger generalized resolution and projectivity than comparable regular designs.
Abstract: The study of good nonregular fractional factorial designs has received significant attention over the last two decades Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs Moreover, some of these designs are found to have maximum projectivity among all designs

3 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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