Minimum Aberration 2 k–p Designs
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.
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TL;DR: In this paper, it was shown that all minimum aberration designs are projections of the design with 5N/16 factors which is constructed by repeatedly doubling the 2 5-1 design defined by I = ABCDE.
Abstract: Given a two-level regular fractional factorial design of resolution IV, the method of doubling produces another design of resolution IV which doubles both the run size and the number of factors of the initial design. On the other hand, the projection of a design of resolution IV onto a subset of factors is of resolution IV or higher. Recent work in the literature of projective geometry essentially determines the structures of all regular designs of resolution IV with n ≥ N/4 + 1 in terms of doubling and projection, where N is the run size and n is the number of factors. These results imply that, for instance, all regular designs of resolution IV with 5N/ 16 < n ≤ N /2 must be projections of the regular design of resolution IV with N/2 factors. We show that, for 9N/32 < n ≤ 5 N/16, all minimum aberration designs are projections of the design with 5N/16 factors which is constructed by repeatedly doubling the 2 5-1 design defined by I = ABCDE. To prove this result, we also derive some properties of doubling, including an identity that relates the wordlength pattern of a design to that of its double and a result that does the same for the alias patterns of two-factor interactions.
56 citations
TL;DR: This paper proposes a new algorithm based on the centered L2-discrepancy, a measure of uniformity, for detecting the isomorphism of fractional factorial designs and shows that the new algorithm is highly reliable and can significantly reduce the complexity of the computation.
54 citations
Cites background from "Minimum Aberration 2 k–p Designs"
...For a detailed discussion refer to Fries and Hunter (1980)....
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TL;DR: In this article, uniformity is used to further distinguish fractional factorial designs, besides the minimum aberration criterion, and optimal designs with 27 and 81 runs are obtained for practical use.
Abstract: The minimum aberration criterion has been frequently used in the selection of fractional factorial designs with nominal factors. For designs with quantitative factors, however, level permutation of factors could alter their geometrical structures and statistical properties. In this paper uniformity is used to further distinguish fractional factorial designs, besides the minimum aberration criterion. We show that minimum aberration designs have low discrepancies on average. An efficient method for constructing uniform minimum aberration designs is proposed and optimal designs with 27 and 81 runs are obtained for practical use. These designs have good uniformity and are effective for studying quantitative factors.
53 citations
Cites background from "Minimum Aberration 2 k–p Designs"
...The minimum aberration criterion [Fries and Hunter (1980)] has been frequently used in the selection of regular fractional factorial (FF) designs with nominal factors, as it provides nice design properties....
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01 Jul 2009
TL;DR: 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.
Abstract: 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.
53 citations
01 Jan 2006
TL;DR: In this paper, the 2 (D) criterion is generalized to the so-called minimum 2 criterion and connections among dieren t criteria are investigated, which provides strong statistical justication for each of them.
Abstract: In recent years, there has been increasing interest in the study of asym- metrical fractional factorial designs. Various new optimality criteria have been proposed from dieren t principles for design construction and comparison, such as generalized minimum aberration, minimum moment aberration, minimum projec- tion uniformity and the 2 (D) (for design D) criteria. In this paper, these criteria are reviewed and the 2 (D) criterion is generalized to the so-called minimum 2 criterion. Connections among dieren t criteria are investigated. These connections provide strong statistical justication for each of them. Some general optimality results are developed, which not only unify several results (including results for the symmetrical case), but also are useful for constructing asymmetrical supersaturated designs.
51 citations
Cites background from "Minimum Aberration 2 k–p Designs"
...Regular FFDs are often constructed to be of minimum aberration (Fries and Hunter (1980)), since this criterion limits the adverse effects of aliasing....
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References
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471 citations
Book•
23 Jun 1976
TL;DR: In conclusion, the size of Industrial Experiments, Fractional Replication--Elementary, and Incomplete Factorials are found to be about the same as that of conventional comparison experiments.
Abstract: Introduction. Simple Comparison Experiments. Two Factors, Each at Two Levels. Two Factors, Each at Three Levels. Unreplicated Three--Factor, Two--Level Experiments. Unreplicated Four--Factor, Two--Level Experiments. Three Five--Factor, Two--Level Unreplicated Experiments. Larger Two--Way Layouts. The Size of Industrial Experiments. Blocking Factorial Experiments, Fractional Replication--Elementary. Fractional Replication--Intermediate. Incomplete Factorials. Sequences of Fractional Replicates. Trend--Robust Plans. Nested Designs. Conclusions and Apologies.
311 citations
09 Jun 1976
271 citations
TL;DR: Incomplete Factorials, Fractional Replication, Intermediate Factorial, and Nested Designs as discussed by the authors are some of the examples of incomplete Factorial Experiments and incomplete fractional replicates.
Abstract: Introduction. Simple Comparison Experiments. Two Factors, Each at Two Levels. Two Factors, Each at Three Levels. Unreplicated Three--Factor, Two--Level Experiments. Unreplicated Four--Factor, Two--Level Experiments. Three Five--Factor, Two--Level Unreplicated Experiments. Larger Two--Way Layouts. The Size of Industrial Experiments. Blocking Factorial Experiments, Fractional Replication--Elementary. Fractional Replication--Intermediate. Incomplete Factorials. Sequences of Fractional Replicates. Trend--Robust Plans. Nested Designs. Conclusions and Apologies.
252 citations