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: This article considers the problem of classifying and ranking designs that are based on Hadamard matrices and finds that generalized aberration performs quite well under these familiar criteria.
Abstract: Deng and Tang (1999) and Tang and Deng (1999) proposed and justified two criteria of generalized minimum aberration for general two-level fractional factorial designs. The criteria are defined using a set of values called J characteristics. In this article, we examine the practical use of the criteria in design selection. Specifically, we consider the problem of classifying and ranking designs that are based on Hadamard matrices. A theoretical result on J characteristics is developed to facilitate the computation. The issue of design selection is further studied by linking generalized aberration with the criteria of efficiency and estimation capacity. Our studies reveal that generalized aberration performs quite well under these familiar criteria.
103 citations
Cites background from "Minimum Aberration 2 k–p Designs"
...Minimum aberration (Fries and Hunter 1980) is the most popular criterion for selecting regular factorials....
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TL;DR: In this paper, the concept of minimum aberration was extended to fractional factorial designs by treating the treatment and blocking factors differently in terms of their contribution to word length in the defining contrast subgroup.
Abstract: Systematic sources of variation can be reduced in fractional factorial experiments by grouping the runs into blocks. This is accomplished through the use of blocking factors. The concepts of resolution and minimum aberration, design optimization criteria ordinarily used to rank unblocked fractional factorial designs, are extended to such blocked fractional factorial designs by treating the treatment and blocking factors differently in terms of their contribution to word length in the defining contrast subgroup. Some limited theoretical results are derived, and tables of minimumaberration blocked two-level fractional factorial designs are presented and considered. The relationship between clear effects (effects that are estimable when higher-order effects are assumed negligible) and minimum aberration in the presence of blocking is discussed.
102 citations
TL;DR: In this paper, the Fries and Hunter algorithm was used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime, and a matrix is given for generating 3 n-m designs with m, n ≤ 6, which have, or nearly have, minimum aberration.
Abstract: Fries and Hunter (1980) presented a practical algorithm for selecting standard 2 n–m fractional factorial designs based on a criterion they called “minimum aberration.” In this article some simple results are presented that enable the Fries–Hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime. Examples of minimum aberration 2 n–m designs with resolution R ≥ 4 are given for m, n – m < 9. A matrix is given for generating 3 n–m designs with m, n – m ≤ 6, which have, or nearly have, minimum aberration.
101 citations
TL;DR: It is demonstrated that any foldover plan of a 2k−p fractional factorial design is equivalent to a core fold over plan consisting only of the p out of k factors, and it is proved that there are exactly 2K−p foldover plans that are equivalent to any core foldoverPlan of a2k−P design.
Abstract: A commonly used follow-up experiment strategy involves the use of a foldover design by reversing the signs of one or more columns of the initial design Defining a foldover plan as the collection of columns whose signs are to be reversed in the foldover design, this article answers the following question: Given a 2k−p design with k factors and p generators, what is its optimal foldover plan? We obtain optimal foldover plans for 16 and 32 runs and tabulate the results for practical use Most of these plans differ from traditional foldover plans that involve reversing the signs of one or all columns There are several equivalent ways to generate a particular foldover design We demonstrate that any foldover plan of a 2k−p fractional factorial design is equivalent to a core foldover plan consisting only of the p out of k factors Furthermore, we prove that there are exactly 2k−p foldover plans that are equivalent to any core foldover plan of a 2k−p design and demonstrate how these foldover plans can be const
96 citations
Cites methods from "Minimum Aberration 2 k–p Designs"
...The criterion that we use is the aberration (Fries and Hunter 1980) of the combined design, as de ned in Section 2....
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Journal Article•
TL;DR: A Bayesian method based on the idea of model discrimination that uncovers the active factors is developed for designing a follow-up experiment to resolve ambiguity in fractional experiments.
Abstract: Fractional factorial, Plackett-Burman, and other multifactor designs are often effective in practice due to factor sparsity. That is, just a few of the many factors studied will have major effects. In those active factors, these designs can have high resolution. We have previously developed a Bayesian method based on the idea of model discrimination that uncovers the active factors. Sometimes, the results of a fractional experiment are ambiguous due to confounding among the possible effects, and more than one model may be consistent with the data. Within the Bayesian construct, we have developed a method for designing a follow-up experiment to resolve this ambiguity. The idea is to choose runs that allow maximum discrimination among the plausible models. This method is more general than methods that algebraically decouple aliased interactions and more appropriate than optimal design methods that require specification of a single model. The method is illustrated through examples of fractional experiments.
90 citations
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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