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Journal Article•DOI•

A graph-aided method for planning two-level experiments when certain interactions are important

Chien-Fu Wu1, Youyi Chen1•
University of Waterloo1
01 May 1992-Technometrics (Taylor & Francis Group)-Vol. 34, Iss: 2, pp 162-175
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.
Citations
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Book Chapter•DOI•
7 Robust design: Experiments for improving quality

[...]

David M. Steinberg
01 Jan 1996-Handbook of Statistics
TL;DR: This chapter discusses the planning, analysis, and interpretation of robust design experiments and the role they play in quality improvement, with special emphasis on issues relating to experimental design.
Abstract: Publisher Summary This chapter discusses the planning, analysis, and interpretation of robust design experiments and the role they play in quality improvement, with special emphasis on issues relating to experimental design. Robust design refers to quality engineering activities whose goal is the development of low-cost, yet high quality products and processes. A key tool in robust design is the use of statistically planned experiments to identify factors that affect product quality and to optimize their nominal levels. The goal of a robust design experiment is to find settings of the design factors that achieve a particular response with high consistency. The most common objectives include (1) maximizing the response, (2) minimizing the response, and (3) keeping the response on target. The use of split plot designs, combined arrays and sequential experimentation will help reduce the size of robust design experiments. Recent initiatives to use response surface methodology in robust design are also a step in the right direction. Robust design experiments can provide the knowledge necessary to make effective design choices by discovering the input factors that affect product quality.

23 citations

Journal Article•
Selecting 2m-p designs using a minimum aberration criterion when some two-factor interactions are importan

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Weiming Ke, Boxin Tang
01 Jan 2004-Quality Engineering
TL;DR: This work proposes and studies a minimum aberration criterion and examines how to search for the best designs according to this criterion, and presents some results for designs of 16 and 32 runs.

23 citations

Cites background or methods from "A graph-aided method for planning t..."

  • ...Much work has been done on nding a design allowing estimation of a set of speci ed effects, often referred to as a requirement set in the literature (see, e.g., Green eld 1976; Franklin 1985; Wu and Chen 1992)....

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  • ...Much work has been done on Ž nding a design allowing estimation of a set of speciŽ ed effects, often referred to as a requirement set in the literature (see, e.g., GreenŽ eld 1976; Franklin 1985; Wu and Chen 1992)....

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  • ...Wu and Chen (1992) suggested using the usual minimum aberration criterion, in combinationwith a graphical method, to select designs when some 2 ’s are important....

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Journal Article•DOI•
Construction of blocked two-level regular designs with general minimum lower order confounding

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Shengli Zhao1, Pengfei Li2, Runchu Zhang3, Runchu Zhang4, Runchu Zhang5, Rohana J. Karunamuni6 •
Qufu Normal University1, University of Waterloo2, Nankai University3, University of British Columbia4, Northeast Normal University5, University of Alberta6
01 Jun 2013-Journal of Statistical Planning and Inference
TL;DR: Wei et al. as discussed by the authors obtained the B1-GMC 2 n − m : 2 r designs with n ≥ 5 N / 16 + 1, where 2 r denotes a two-level regular blocked design with n = 2 n−m runs, n treatment factors, and 2 r blocks.

22 citations

Additional excerpts

  • ...The second is based on the clear effects criterion (Wu and Chen, 1992)....

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Book•
Advanced experimental design

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Klaus Hinkelmann, Oscar Kempthorne
01 Jan 2005
TL;DR: In this paper, the authors present an analysis of incomplete block designs with respect to the following properties: 1.1 Introduction. 2.2 Definition of BIB design. 3.3 Parameterization in terms of main effects and interactions. 4.4 Incomplete block design with variable block size.
Abstract: Preface. 1. General Incomplete Block Design. 1.1 Introduction and Examples. 1.2 General Remarks on the Analysis of Incomplete Block Designs. 1.3 The Intrablock Analysis. 1.4 Incomplete Designs with Variable Block Size. 1.5 Disconnected Incomplete Block Designs. 1.6 Randomization Analysis. 1.7 Interblock Information in an Incomplete Block Design. 1.8 Combined Intra- and Interblock Analysis. 1.9 Relationships Among Intrablock, Interblock, and Combined Estimation. 1.10 Estimation of Weights for the Combined Analysis. 1.11 Maximum-Likelihood Type Estimation. 1.12 Efficiency Factor of an Incomplete Block Design. 1.13 Optimal Designs. 1.14 Computational Procedures. 2. Balanced Incomplete Block Designs. 2.1 Introduction. 2.2 Definition of the BIB Design. 2.3 Properties of BIB Designs. 2.4 Analysis of BIB Designs. 2.5 Estimation of rho. 2.6 Significance Tests. 2.7 Some Special Arrangements. 2.8 Resistant and Susceptible BIB Designs. 3. Construction of Balanced Incomplete Block Designs. 3.1 Introduction. 3.2 Difference Methods. 3.3 Other Methods. 3.4 Listing of Existing BIB Designs. 4. Partially Balanced Incomplete Block Designs. 4.1 Introduction. 4.2 Preliminaries. 4.3 Definition and Properties of PBIB Designs. 4.4 Association Schemes and Linear Associative Algebras. 4.5 Analysis of PBIB Designs. 4.6 Classification of PBIB Designs. 4.7 Estimation of rho for PBIB(2) Designs. 5. Construction of Partially Balanced Incomplete Block Designs. 5.1 Group-Divisible PBIB(2) Designs. 5.2 Construction of Other PBIB(2) Designs. 5.3 Cyclic PBIB Designs. 5.4 Kronecker Product Designs. 5.5 Extended Group-Divisible PBIB Designs. 5.6 Hypercubic PBIB Designs. 6. More Block Designs and Blocking Structures. 6.1 Introduction. 6.2 Alpha Designs. 6.3 Generalized Cyclic Incomplete Block Designs. 6.4 Designs Based on the Successive Diagonalizing Method. 6.5 Comparing Treatments with a Control. 6.6 Row-Column Designs. 7. Two-Level Factorial Designs. 7.1 Introduction. 7.2 Case of Two Factors. 7.3 Case of Three Factors. 7.4 General Case. 7.5 Interpretation of Effects and Interactions. 7.6 Analysis of Factorial Experiments. 8. Confounding in 2n Factorial Designs. 8.1 Introduction. 8.2 Systems of Confounding. 8.3 Composition of Blocks for a Particular System of Confounding. 8.4 Detecting a System of Confounding. 8.5 Using SAS for Constructing Systems of Confounding. 8.6 Analysis of Experiments with Confounding. 8.7 Interblock Information in Confounded Experiments. 8.8 Numerical Example Using SAS. 9. Partial Confounding in 2n Factorial Designs. 9.1 Introduction. 9.2 Simple Case of Partial Confounding. 9.3 Partial Confounding as an Incomplete Block Design. 9.4 Efficiency of Partial Confounding. 9.5 Partial Confounding in a 23 Experiment. 9.6 Partial Confounding in a 24 Experiment. 9.7 General Case. 9.7.1 Intrablock Information. 9.8 Double Confounding. 9.9 Confounding in Squares. 9.10 Numerical Examples Using SAS. 10. Designs with Factors at Three Levels. 10.1 Introduction. 10.2 Definition of Main Effects and Interactions. 10.3 Parameterization in Terms of Main Effects and Interactions. 10.4 Analysis of 3n Experiments. 10.5 Confounding in a 3n Factorial. 10.6 Useful Systems of Confounding. 10.7 Analysis of Confounded 3n Factorials. 10.8 Numerical Example. 11. General Symmetrical Factorial Design. 11.1 Introduction. 11.2 Representation of Effects and Interactions. 11.3 Generalized Interactions. 11.4 Systems of Confounding. 11.5 Intrablock Subgroup. 11.6 Enumerating Systems of Confounding. 11.7 Fisher Plans. 11.8 Symmetrical Factorials and Finite Geometries. 11.9 Parameterization of Treatment Responses. 11.10 Analysis of pn Factorial Experiments. 11.11 Interblock Analysis. 11.12 Combined Intra- and Interblock Information. 11.13 The sn Factorial. 11.14 General Method of Confounding for the Symmetrical Factorial Experiment. 11.15 Choice of Initial Block. 12. Confounding in Asymmetrical Factorial Designs. 12.1 Introduction. 12.2 Combining Symmetrical Systems of Confounding. 12.3 The GC/n Method. 12.4 Method of Finite Rings. 12.5 Balanced Factorial Designs (BFD). 13. Fractional Factorial Designs. 13.1 Introduction. 13.2 Simple Example of Fractional Replication. 13.3 Fractional Replicates for 2n Factorial Designs. 13.4 Fractional Replicates for 3n Factorial Designs. 13.5 General Case of Fractional Replication. 13.6 Characterization of Fractional Factorial Designs of Resolution III, IV, and V. 13.7 Fractional Factorials and Combinatorial Arrays. 13.8 Blocking in Fractional Factorials. 13.9 Analysis of Unreplicated Factorials. 14. Main Effect Plans. 14.1 Introduction. 14.2 Orthogonal Resolution III Designs for Symmetrical Factorials. 14.3 Orthogonal Resolution III Designs for Asymmetrical Factorials. 14.4 Nonorthogonal Resolution III Designs. 15. Supersaturated Designs. 15.1 Introduction and Rationale. 15.2 Random Balance Designs. 15.3 Definition and Properties of Supersaturated Designs. 15.4 Construction of Two-Level Supersaturated Designs. 15.5 Three-Level Supersaturated Designs. 15.6 Analysis of Supersaturated Experiments. 16. Search Designs. 16.1 Introduction and Rationale. 16.2 Definition of Search Design. 16.3 Properties of Search Designs. 16.4 Listing of Search Designs. 16.5 Analysis of Search Experiments. 16.6 Search Probabilities. 17. Robust-Design Experiments. 17.1 Off-Line Quality Control. 17.2 Design and Noise Factors. 17.3 Measuring Loss. 17.4 Robust-Design Experiments. 17.5 Modeling of Data. 18. Lattice Designs. 18.1 Definition of Quasi-Factorial Designs. 18.2 Types of Lattice Designs. 18.3 Construction of One-Restrictional Lattice Designs. 18.4 General Method of Analysis for One-Restrictional Lattice Designs. 18.5 Effects of Inaccuracies in the Weights. 18.6 Analysis of Lattice Designs as Randomized Complete Block Designs. 18.7 Lattice Designs as Partially Balanced Incomplete Block Designs. 18.8 Lattice Designs with Blocks of Size Kl. 18.9 Two-Restrictional Lattices. 18.10 Lattice Rectangles. 18.11 Rectangular Lattices. 18.12 Efficiency Factors. 19. Crossover Designs. 19.1 Introduction. 19.2 Residual Effects. 19.3 The Model. 19.4 Properties of Crossover Designs. 19.5 Construction of Crossover Designs. 19.6 Optimal Designs. 19.7 Analysis of Crossover Designs. 19.8 Comments on Other Models. Appendix A: Fields and Galois Fields. Appendix B: Finite Geometries. Appendix C: Orthogonal and Balanced Arrays. Appendix D: Selected Asymmetrical Balanced Factorial Designs. Appendix E: Exercises. References. Author Index. Subject Index.

22 citations

Journal Article•DOI•
Some theory for constructing general minimum lower order confounding designs

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Jie Chen, Min-Qian Liu
01 Oct 2011-Statistica Sinica
TL;DR: In this paper, the authors show that all GMC designs with n runs and n two-level factors are projections of maximal designs with N/2, 5N/16, and 9N/32 factors, respectively.
Abstract: General minimum lower order confounding (GMC) is a newly proposed design criterion that aims at keeping the lower order effects unaliased with one another to the extent possible. This paper shows that for 5N/16 < n ≤ N/2, 9N/32 < n ≤ 5N/16, and 17N/64 < n ≤ 9N/32, all GMC designs with N runs and n two-level factors are projections of maximal designs with N/2, 5N/16, and 9N/32 factors, respectively. Furthermore, it provides immediate approaches to construct- ing these GMC designs from the respective maximal designs; these approaches can produce many more GMC designs than the existing computer search method.

19 citations

Cites background from "A graph-aided method for planning t..."

  • ...Among them, the minimum aberration (MA) criterion introduced by Fries and Hunter (1980) that treats factorial effects of the same order as equally important and lower order effects as more important than higher order ones, and the clear effects criterion of Wu and Chen (1992) that concerns effects not aliased with main effects and two-factor interactions. Recently, Zhang et al. (2008) introduced the aliased effect-number pattern and, based on this pattern, they proposed a new criterion of general minimum lower order confounding (GMC) that aims at keeping the lower order effects unaliased with one another in an explicit manner....

    [...]

  • ...Among them, the minimum aberration (MA) criterion introduced by Fries and Hunter (1980) that treats factorial effects of the same order as equally important and lower order effects as more important than higher order ones, and the clear effects criterion of Wu and Chen (1992) that concerns effects not aliased with main effects and two-factor interactions....

    [...]

  • ...Among them, the minimum aberration (MA) criterion introduced by Fries and Hunter (1980) that treats factorial effects of the same order as equally important and lower order effects as more important than higher order ones, and the clear effects criterion of Wu and Chen (1992) that concerns effects not aliased with main effects and two-factor interactions. Recently, Zhang et al. (2008) introduced the aliased effect-number pattern and, based on this pattern, they proposed a new criterion of general minimum lower order confounding (GMC) that aims at keeping the lower order effects unaliased with one another in an explicit manner. Zhang and Mukerjee (2009) characterized the GMC criterion via complementary sets and listed the complementary designs of some GMC designs....

    [...]

  • ...Among them, the minimum aberration (MA) criterion introduced by Fries and Hunter (1980) that treats factorial effects of the same order as equally important and lower order effects as more important than higher order ones, and the clear effects criterion of Wu and Chen (1992) that concerns effects not aliased with main effects and two-factor interactions. Recently, Zhang et al. (2008) introduced the aliased effect-number pattern and, based on this pattern, they proposed a new criterion of general minimum lower order confounding (GMC) that aims at keeping the lower order effects unaliased with one another in an explicit manner. Zhang and Mukerjee (2009) characterized the GMC criterion via complementary sets and listed the complementary designs of some GMC designs. There has been some work on constructing GMC designs. Zhang et al. (2008) used computer search when run size is small, but this approach is time consuming and even inefficient when the run size is as large as 128....

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  • ...…by Fries and Hunter (1980) that treats factorial effects of the same order as equally important and lower order effects as more important than higher order ones, and the clear effects criterion of Wu and Chen (1992) that concerns effects not aliased with main effects and two-factor interactions....

    [...]

References
More filters
Book•
Statistics for experimenters

[...]

Sidney Addelman
01 Jan 1978

5,151 citations

Book•
Quality Engineering Using Robust Design

[...]

Madhan Shridhar Phadke1•
Hewlett-Packard1
01 Jan 2015
TL;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)....

    [...]

  • ...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)....

    [...]

Journal Article•DOI•
The graph isomorphism disease

[...]

Ronald C. Read1, Derek G. Corneil2•
University of Waterloo1, University of Toronto2
01 Dec 1977-Journal of Graph Theory
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

Journal Article•DOI•
Minimum Aberration 2 k–p Designs

[...]

Arthur Fries, William G. Hunter
01 Nov 1980-Technometrics
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

Minimum Aberration 2k-p Designs

[...]

Arthur Fries, William G. Hunter
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

Related Papers (5)
Minimum Aberration 2 k–p Designs

[...]

01 Nov 1980-Technometrics
Arthur Fries, William G. Hunter
Minimum Aberration 2k-p Designs

[...]

01 Jan 1980
Arthur Fries, William G. Hunter
A catalogue of two-level and three-level fractional factorial designs with small runs

[...]

01 Apr 1993-International Statistical Review
Jiahua Chen, D. X. Sun, C. F. J. Wu
Experiments: Planning, Analysis, and Parameter Design Optimization

[...]

01 Jan 2000
C. F. Jeff Wu, Michael S. Hamada
The 2 k — p Fractional Factorial Designs

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01 Aug 1961-Technometrics
George E. P. Box, J. S. Hunter