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Journal ArticleDOI

Finding New Fractions of Factorial Experimental Designs

01 Aug 1961-Technometrics (Taylor & Francis Group)-Vol. 3, Iss: 3, pp 359-370
TL;DR: In this paper, a method of obtaining symmetrical balanced fractions of 3 n and 2 m 3 n factorial designs is proposed, based on an analysis of such designs into a complex of concentric hyperspheres in an n-dimensional factor space.
Abstract: A method of obtaining symmetrical balanced fractions of 3 n and 2 m 3 n factorial designs is proposed, based on an analysis of such designs into a complex of concentric hyperspheres in an n-dimensional factor space. Two examples are constructed, a half-replicate of a 34 design and a half-replicate of a 23 32 design. Analysis shows both designs to have useful properties and to be relatively easy to analyse. Comparison is made with a half-replicate of a 23 32 design recently published by W. S. Connor.
Citations
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Journal ArticleDOI
TL;DR: In this paper, D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models ( 2 ≤ p ≥ 5) using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimally balanced array designs.
Abstract: D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.

47 citations

Journal ArticleDOI
TL;DR: In this paper, a review of techniques for obtaining the treatment combinations that comprise a fraction of a factorial arrangement is presented, which includes orthogonal and non-orthogonal plans for both symmetrical and asymmetrical factorial experiments.
Abstract: This paper is a review of techniques for obtaining the treatment combinations that comprise a fraction of a factorial arrangement. Several procedures for constructing fractional replicate plans, which in their original form appear to be different, are presented in a manner which illustrates their similarities. The techniques discussed include orthogonal and non-orthogonal plans for both symmetrical and asymmetrical factorial experiments. The plans developed range from those that permit estimation of main effects only, to those that permit estimation of main effects and all two-factor interaction effects.

45 citations

01 May 1966
TL;DR: In this paper, the authors consider the problem when there are some factors at two levels and some at three levels, and they show that a restricted model and special experimental designs are needed.
Abstract: : The choice of an experimental design suitable for fitting a graduating polynomial can be made according to a number of criteria, depending on the problem involved. Difficulties arise when, although the factors are continuous in nature, the number of levels is specified by some external considerations. For example, if some factors can be examined at only two levels, the graduating function cannot include quadratic terms in those variables, but all second order terms for variables to be examined at three or more levels can be permitted. For such cases, a restricted model and special experimental designs are needed. This paper considers the problem when there are some factors at two levels and some factors at three levels. (Author)

13 citations


Cites background from "Finding New Fractions of Factorial ..."

  • ...A further property of the 432 design points is that they can be divided into the four groups A, B, C, and D defined above in such a way that the points of each group lie on a hypersphere in seven dimensional space (see Fry, 1961)....

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Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem when there are some factors at two levels, some at three levels, and some at four levels, where the number of levels is specified by some external considerations.
Abstract: The choice of an experimental design suitable for fitting a graduating polynomial can be made according to a number of criteria, depending on the problem involved. Difficulties arise when, although the factors are continuous in nature, the number of levels is specified by some external considerations. For example, if some factors can be examined at only two levels, the graduating function cannot include quadratic terms in those variables, but all second order terms for variables to be examined at three or more levels can be permitted. For such cases, a restricted model and special experimental designs are needed. This paper considers the problem when there are some factors at two levels and some factors at three levels, and when there are some factors at two levels and some factors at four levels.

12 citations

Journal ArticleDOI
TL;DR: In this paper, three construction techniques are discussed which yield designs providing orthogonal estimates of all the main effects and allowing estimation of all two-factor interactions for the 2n3m factorial series.
Abstract: If we assume no higher order interactions for the 2n3m factorial series of designs, then relaxing the restrictions concerning equal frequency for the factors and complete orthogonality for each estimate permits considerable savings in the number of runs required to estimate all the main effects and two-factor interactions. Three construction techniques are discussed which yield designs providing orthogonal estimates of all the main effects and allowing estimation of all the two-factor interactions. These techniques are: (i) collapsing of factors in symmetrical fractionated 3m–p designs, (ii) conjoining fractionated designs, and (iii) combinations of (i) and (ii). Collapsing factors in a design either maintains or increases the resolution of the original design, but does not decrease it. Plans are presented for certain values of (n, m) as examples of the construction techniques. Systematic methods of analysis are also discussed.

11 citations


Cites methods from "Finding New Fractions of Factorial ..."

  • ...Another method of construction of incomplete 2n3m designs was presented by Fry [15], yet this technique yielded designs quite similar to those produced by Connor and Young....

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References
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Journal ArticleDOI
TL;DR: In this paper, a class of incomplete three level factorial designs useful for estimating the coefficients in a second degree graduating polynomial are described and the designs either meet, or approximately meet, the criterion of rotatability and for the most part can be orthogonally blocked.
Abstract: A class of incomplete three level factorial designs useful for estimating the coefficients in a second degree graduating polynomial are described. The designs either meet, or approximately meet, the criterion of rotatability and for the most part can be orthogonally blocked. A fully worked example is included.

3,194 citations


"Finding New Fractions of Factorial ..." refers background in this paper

  • ...In an interesting paper by Box and Behnken [10], some fractions of 3" designs are derived by an ingenious use of the properties of balanced incomplete block designs....

    [...]

Book
15 Jan 1952
TL;DR: In this article, Monterey describes a books design and analysis of experiments, and the pronouncement as without difficulty as perspicacity of this design and analyses of experiments montgomery can be taken as skillfully as picked to act.
Abstract: Yeah, reviewing a books design and analysis of experiments montgomery could mount up your close associates listings. This is just one of the solutions for you to be successful. As understood, achievement does not suggest that you have wonderful points. Comprehending as skillfully as covenant even more than extra will have enough money each success. next-door to, the pronouncement as without difficulty as perspicacity of this design and analysis of experiments montgomery can be taken as skillfully as picked to act. Page Url

1,064 citations

Journal ArticleDOI
TL;DR: Oscar Kempthorne as discussed by the authors, The Design and Analysis of Experiments. New York: John Wiley and Sons; London: Chapman and Hall, 1952. Pp. xix + 631.
Abstract: Oscar Kempthorne: The Design and Analysis of Experiments. New York: John Wiley and Sons; London: Chapman and Hall, 1952. Pp. xix + 631. 68s.

381 citations

Journal ArticleDOI
TL;DR: The topics to be covered include the following: 1.
Abstract: Random balance experimental design has been used in industrial applications of statistical methods since 1956. The purpose of this report is to record and discuss in specific form key points regarding this technique. This report will be divided into several parts that are essentially separate. The topics to be covered include the following: 1. Random design and the motivations for use of random designs in industrial, engineering and research activities. 2. Techniques for analysis of random data and the apparent application domains for which each technique is appropriate. 3. Fundamental problems of statistical analysis that are of critical importance in some random balance applications.

266 citations