A graph-aided method for planning two-level experiments when certain interactions are important
TL;DR: In this paper, a graph-aided method is proposed to solve the problem of fractional factorial factorial experiment planning, where prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects.
Abstract: In planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects. In this article, we propose a graph-aided method to solve this problem for two-level experiments. First, we choose the defining relations for a 2 n–k design according to a goodness criterion such as the minimum aberration criterion. Then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two-factor interactions for the given defining relations. In each graph a vertex represents a factor and an edge represents the interaction between the two factors. For the experiment planner, the job is simple: Draw a graph representing the specified interactions and compare it with the list of graphs obtained previously. Our approach is a substantial improvement over Taguchi's linear graphs.
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TL;DR: Zhang et al. as mentioned in this paper constructed GMC 2n41 designs with 5N/16 + 1 ≤ n + 2 < N − 1, where n and N are respectively the numbers of two-level factors and runs.
Abstract: General minimum lower-order confounding (GMC) criterion is to choose optimal designs, which is based on the aliased effect-number pattern (AENP). The AENP and GMC criterion have been developed to form GMC theory. Zhang, Yang, Li and Zhang (2015) introduced GMC 2n4m criterion for choosing optimal designs and constructed all GMC 2n41 designs with N/4 + 1 ≤ n + 2 ≤ 5N/16. In this paper, we analyze properties of 2n41 designs and construct GMC 2n41 designs with 5N/16 + 1 ≤ n + 2 < N − 1, where n and N are respectively the numbers of two-level factors and runs. Further, GMC 2n41 designs with 16-run, 32-run are tabulated.
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Cites background from "A graph-aided method for planning t..."
...…to compare fractions, such as maximum resolution (MR) (Box and Hunter, 1961), minimum aberration (MA) (Fries and Hunter, 1980), clear effects (CE) (Wu and Chen, 1992), maximum estimation capacity (MEC) (Sun, 1993), and general minimum lower-order confounding (GMC) (Zhang et al., 2008) criteria....
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TL;DR: In this paper, the general minimum lower order confounding (GMC) criterion was proposed to select factorial designs, called GMC designs, and the theory of constructing GMC 2n-m designs with 5N/16+1≤n≤N−1 was studied.
Abstract: The general minimum lower order confounding (GMC) criterion was proposed to select factorial designs, called GMC designs. The theory of constructing GMC 2n–m designs with 5N/16+1≤n≤N−1 was studied ...
1 citations
Cites background from "A graph-aided method for planning t..."
...A 2FI is called clear if it is not aliased with any ME’s or other 2FI’s (Wu and Chen 1992)....
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...A 2FI is called clear if it is not aliased with any ME’s or other 2FI’s (Wu and Chen 1992). The clear ME’s and 2FI’s can be estimated without the contamination of other ME’s and 2FI’s. Focusing on the interactions between control and noise factors, Ke, Tang, and Wu (2005) introduced four classes of clear compromise plans, in which the columns are divided into two groups G1 and G2....
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TL;DR: In this paper, the coset pattern matrix is used to characterize minimum M -aberration designs through their complements, and an identity is established to characterize a design through its complement.
Abstract: The coset pattern matrix contains more detailed information about effect aliasing in a factorial design than the commonly used wordlength pattern. More flexible and elaborate design criteria can be proposed using the coset pattern matrix. In this article, we establish an identity that relates the coset pattern matrix of a design to that of its complement. As an application, the identity is used to characterize minimum M -aberration designs through their complements.
1 citations
TL;DR: In this paper, a unified approach to the construction of regular factorial designs with randomization restrictions using randomization defining contrast subspaces for the representation of randomization restriction is presented.
Abstract: Regular factorial designs with randomization restrictions are widely used in practice. This paper provides a unified approach to the construction of such designs using randomization defining contrast subspaces for the representation of randomization restrictions. We use finite projective geometry to determine the existence of designs with the required structure and develop a systematic approach for their construction. An attractive feature is that commonly used factorial designs with randomization restrictions are special cases of this general representation. Issues related to the use of these designs for particular factorial experiments are also addressed.
1 citations
11 Oct 2015
TL;DR: In this paper, the authors considered a model involving the main plus two-factor interaction effects with their interest lying in the estimation of the main effects and a specified set of two factor interaction effects.
Abstract: Over two decades, optimal choice designs have been obtained for estimating the main effects and the main plus two-factor interaction effects under both the multinomial logit model and the linear paired comparison model. However, there are no general results on the optimal choice designs for estimating main plus some two-factor interaction effects. We consider a model involving the main plus two-factor interaction effects with our interest lying in the estimation of the main effects and a specified set of two-factor interaction effects. The specified set of the two-factor interaction effects include the interactions where only one of the factors possibly interact with the other factors. We first characterize the information matrix and then construct universally optimal choice designs for choice set sizes 3 and 4.
Additional excerpts
...In the traditional factorial design setup, the issue of estimability and optimality in situations of this kind has been addressed by Hedayat and Pesotan (1992), Wu and Chen (1992), Hedayat and Pesotan (1997), Chiu and John (1998), Dey and Mukerjee (1999) and Dey and Suen (2002)....
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References
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Book•
01 Jan 2015TL;DR: This book offers a complete blueprint for structuring projects to achieve rapid completion with high engineering productivity during the research and development phase to ensure that high quality products can be made quickly and at the lowest possible cost.
Abstract: From the Publisher:
Phadke was trained in robust design techniques by Genichi Taguchi, the mastermind behind Japanese quality manufacturing technologies and the father of Japanese quality control. Taguchi's approach is currently under consideration to be adopted as a student protocol with the US govrnment. The foreword is written by Taguchi. This book offers a complete blueprint for structuring projects to achieve rapid completion with high engineering productivity during the research and development phase to ensure that high quality products can be made quickly and at the lowest possible cost. Some topics covered are: orthogonol arrays, how to construct orthogonal arrays, computer-aided robutst design techniques, dynamic systems design methods, and more.
3,928 citations
"A graph-aided method for planning t..." refers methods in this paper
...For a review on efficient algorithms for testing graph isomorphism, see Read and Corneil (1977) and Hoffman (1982)....
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...For simplicity we do not include in this article the column numbers for the lines, which can be easily read from the interaction tables given by Phadke (1989) and Taguchi (1987)....
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TL;DR: The present state of the art of isomorphism testing is surveyed, its relationship to NP-completeness is discussed, and some of the difficulties inherent in this particularly elusive and challenging problem are indicated.
Abstract: The graph isomorphism problem—to devise a good algorithm for determining if two graphs are isomorphic—is of considerable practical importance, and is also of theoretical interest due to its relationship to the concept of NP-completeness. No efficient (i.e., polynomial-bound) algorithm for graph isomorphism is known, and it has been conjectured that no such algorithm can exist. Many papers on the subject have appeared, but progress has been slight; in fact, the intractable nature of the problem and the way that many graph theorists have been led to devote much time to it, recall those aspects of the four-color conjecture which prompted Harary to rechristen it the “four-color disease.” This paper surveys the present state of the art of isomorphism testing, discusses its relationship to NP-completeness, and indicates some of the difficulties inherent in this particularly elusive and challenging problem. A comprehensive bibliography of papers relating to the graph isomorphism problem is given.
519 citations
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
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