Minimum Aberration 2 k-p Designs

Fries, Arthur; Hunter, William G.

doi:10.2307/1268198

Cited by 272 publications

(157 citation statements)

References 12 publications

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“…Such a refinement is analogous to using minimum aberration [44] in factorial designs. These designs are the duals of linear codes.…”

Section: Other Kinds Of Optimalitymentioning

confidence: 99%

Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

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To a combinatorialist, a design is usually a 2-design or balanced incompleteblock design. However, 2-designs do not necessarily exist in all cases where a statistician might wish to use one to design an experiment. As a result, statisticians need to consider structures much more general than the combinatorialist's designs, and to decide which one is "best" in a given situation. This leads to the theory of optimal design. There are several concepts of optimality, and no general consensus about which one to use in any particular situation.For block designs with fixed block size k, all these optimality criteria are determined by a graph, the concurrence graph of the design, and more specifically, by the eigenvalues of the Laplacian matrix of the graph. It turns out that the optimality criteria most used by statisticians correspond to properties of this graph which are interesting in other contexts: D-optimality involves maximizing the number of spanning trees; A-optimality involves minimizing the sum of resistances between all pairs of terminals (when the graph is regarded as an electrical circuit, with each edge being a one-ohm resistor); and E-optimality involves maximizing the smallest eigenvalue of the Laplacian (the corresponding graphs are likely to have good expansion and random walk properties). If you are familiar with these properties, you may expect that related "nice" properties such as regularity and large girth (or even symmetry) may tend to hold; some of our examples may come as a surprise! The aim of this paper is to point out that the optimal design point of view unifies various topics in graph theory and design theory, and suggests some interesting open problems to which combinatorialists of all kinds might turn their expertise. We describe in some detail both the statistical background and the mathematics of various topics such as Laplace eigenvalues of graphs. PreliminariesThis first section of the paper sets the scene. We look briefly at the different ways in which a combinatorialist and a statistician view a simple block design such as the Fano plane, and also introduce a running example which would not be recognised as a design under the standard combinatorial definition but is in fact the best design for a statistician in certain circumstances.Section 2 looks at the way that information about treatment differences is recovered from the results of an experiment conducted using a block design. We look briefly at what makes a good design, and show that all criteria for this can be expressed in terms of the concurrence graph of the design. Having thus focussed our attention on graphs, we give a brief survey of the Laplacian matrix of a graph and its eigenvalues, and mention the interesting question of which graphs can be concurrence graphs of block designs with given block size.Section 3 covers the three most important kinds of optimality, described by the letters A (average), D (determinant) and E (extreme), and how they look when expressed in terms of the concurrence matrix.In section ...

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“…Such a refinement is analogous to using minimum aberration [44] in factorial designs. These designs are the duals of linear codes.…”

Section: Other Kinds Of Optimalitymentioning

confidence: 99%

Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

View full text Add to dashboard Cite

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“…The resolution criterion proposed by Box and Hunter (1961) selects 2 m-p designs that have highest resolution. Since two designs having the same resolution may have different word-length patterns and may not be equally good, Fries and Hunter (1980) Suppose we wish to perform an experiment with 8 runs and several factors at two levels, labeled +1 and −1. Table 1 gives the columns of an 8-run saturated design with its columns arranged in Yates' order, with the generating independent columns 1, 2, and 4, in boldface.…”

Section: Two-level Fractional Factorial Designsmentioning

confidence: 99%

“…Since the fraction can be chosen in many different ways, a key concern is how to choose a fraction of the full factorial design for a given run size and the number of factors. Commonly used criterion for 2 m-p design selection is the minimum aberration criterion proposed by Fries and Hunter (1980). For a small number of factors, Box, Hunter, & Hunter (1978) provided a useful catalogue of 2 m-p designs with minimum aberration.…”

Section: Introductionmentioning

confidence: 99%

Selection of Blocked Two-Level Fractional Factorial Designs for Agricultural Experiments

Ke¹,

Ren²,

Lu³

2007

Conference on Applied Statistics in Agriculture

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Blocked two-level fractional factorial designs are a very useful tool for efficient data collection in agricultural and other scientific research. In most experiments, in addition to the main effects, some two-factor interactions are also meaningful and need to be estimated. We propose a method for efficiently selecting blocked two-level fractional factorial designs when some of the two-factor interactions are non-negligible. We then present some results for a design with only 8 or 16 runs to illustrate how to use this method.

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“…Fries and Hunter (1980) proposed a minimum aberration criterion to further distinguish two designs with same resolution by using the concept of the word-length pattern. The run size of a regular design must be a power of two and the defining relations of the regular design completely specify the aliasing pattern.…”

Section: Design Selecting Criteriamentioning

confidence: 99%

Another Look at New Gma Orthogonal Arrays

Li¹,

Wittig²

2001

Conference on Applied Statistics in Agriculture

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Non-regular factorial designs have not been advocated until last decade clue to their complex aliasing structure. However, some researchers recently found that the complex aliasing structure of non-regular factorial designs is a challenge as well as an opportunity. Li, Deng, and Tang (2000) studied nOll-regular designs and generated a collection of non-equivalent orthogonal arrays using a generalized miniumm aberration criterion, proposed by Deng and Tang (1999). Some new orthogonal arrays they found cannot be embedded into Hadamard matrices. In this paper, we study these orthogonal arrays from the angle of projection. vVe show that these new GMA orthogonal arrays are also superior to the top designs obtained from Hadamard matrices when evaluated hy the criteria of model estimability and design efhc:ienc:y.}(e:1J WOTYiS and phmses: non-regular design, gcncrajizccl minimum aberratiun, model cstimaIJility, design efficiency, Hadamard matrices, orthogona.l alTays.

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Minimum Aberration 2 k-p Designs

Cited by 272 publications

References 12 publications

Combinatorics of optimal designs

Combinatorics of optimal designs

Selection of Blocked Two-Level Fractional Factorial Designs for Agricultural Experiments

Another Look at New Gma Orthogonal Arrays

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