Optimality of Certain Asymmetrical Experimental Designs

Cheng, Ching‐Shui

doi:10.1214/aos/1176344371

Cited by 161 publications

(52 citation statements)

References 9 publications

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“…The name arises because its concurrence graph consists of the λ-fold complete graph together with a simple regular graph. Now Chêng [30,31] proved:…”

Section: General Resultsmentioning

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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“…The name arises because its concurrence graph consists of the λ-fold complete graph together with a simple regular graph. Now Chêng [30,31] proved:…”

Section: General Resultsmentioning

confidence: 99%

Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

View full text Add to dashboard Cite

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“…In practice, full factorial designs are impossible for microarrays because only one sample can correspond to each array-dye combination. However it is possible to derive ef cient designs that satisfy the constraints imposed by this technology (John and Mitchell, 1977;Cheng, 1978). In general, for a given number of arrays, designs that are balanced across the samples of interest will provide the greatest ef ciency.…”

Section: Discussionmentioning

confidence: 99%

Analysis of Variance for Gene Expression Microarray Data

Kerr

Martin

Churchill

2000

Journal of Computational Biology

1,154

849

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Spotted cDNA microarrays are emerging as a powerful and cost-effective tool for largescale analysis of gene expression. Microarrays can be used to measure the relative quantities of speci c mRNAs in two or more tissue samples for thousands of genes simultaneously. While the power of this technology has been recognized, many open questions remain about appropriate analysis of microarray data. One question is how to make valid estimates of the relative expression for genes that are not biased by ancillary sources of variation. Recognizing that there is inherent "noise" in microarray data, how does one estimate the error variation associated with an estimated change in expression, i.e., how does one construct the error bars? We demonstrate that ANOVA methods can be used to normalize microarray data and provide estimates of changes in gene expression that are corrected for potential confounding effects. This approach establishes a framework for the general analysis and interpretation of microarray data.

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“…The optimal designs in univariate model in the block design, model (5), were proposed among others by Cheng (1978Cheng ( , 1980, Hedayat and Li (1979), Gaffke (1982), Majumar and Notz (1983), John (1987), Azais et al (1993), Srivastav and Shankar (2003) and the optimal choice of time points in model (4) was determined by Abt et al (1998), Imhof (1998), Gaffke and Heiligers (1998) Hedayat and Li 1979) is D-optimal in model (1) Srivastav and Shankar 2003) is E-optimal in model (1) …”

Section: Is E-optimal (χ E -Optimal) For the Estimation Of B 1 In Modmentioning

confidence: 99%