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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28 Jun 2017
TL;DR: In this paper, the authors derived a formula for computing the (generalized) word length pattern of a union of OAs that makes use of their polynomial counting functions, and the best OAs according to the generalized minimum Aberration criterion can thereby be found simply by exploring a relatively small set of counting functions.
Abstract: We generate all the Orthogonal Arrays (OAs) of a given size n and strength t as the union of a collection of OAs which belong to an inclusion-minimal set of OAs. We derive a formula for computing the (Generalized) Word Length Pattern of a union of OAs that makes use of their polynomial counting functions. The best OAs according to the Generalized Minimum Aberration criterion can thereby be found simply by exploring a relatively small set of counting functions. The classes of OAs with 5 binary factors, strength 2, and sizes 16 and 20 are fully described.
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TL;DR: In this article, the equivalence of the generalized minimum aberration and the minimum χ 2 criteria is reported, which not only provides another justification for each other but also develops some theoretical results for designs with generalized minimum abnormality and some lower bounds for the generalized wordlength pattern.
Abstract: Various optimal criteria have been proposed to rank asymmetrical frac- tional factorial designs. Among them, the generalized minimum aberration and the minimum moment aberration criteria are the most popular ones and have received much attention. Recently, Liu et al. (Stat Sin 16:1285-1297, 2006) proposed the mini- mum χ 2 criterion in terms of level-combinations. In this paper, the equivalency of the generalized minimum aberration and the minimum χ 2 criteria is reported, which not only provides another justification for each other but also develops some theoretical results for designs with generalized minimum aberration and some lower bounds for the generalized wordlength pattern. Besides, an analytic relationship between gener- alized minimum aberration and minimum moment aberration is obtained for asym- metrical fractional factorial designs.
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TL;DR: In this paper, a simple method for the calculation of aliased effect number pattern (AENP) was proposed, which is much easier than before since the calculation only makes use of the design matrix, and all 128 run GMC designs with the number of factors ranging from 8 to 32 are provided for practical use.
Abstract: Motivated by the effect hierarchy principle, Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect number pattern (AENP) for regular fractional factorial designs and based on the new pattern proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal $$2^{n-m}$$
designs. Zhang et al. (Stat Sinica 18:1689–1705, 2008) proved that most existing criteria can be obtained by functions of the AENP. In this paper we propose a simple method for the calculation of AENP. The method is much easier than before since the calculation only makes use of the design matrix. All 128-run GMC designs with the number of factors ranging from 8 to 32 are provided for practical use.
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TL;DR: In this article, all the inequivalent projection designs from Hadamard matrices of order 32, 36 and 40 into q = 3, 4, 5 factors and order 32 into 6 factors were compared.
2 citations
10 May 2005
TL;DR: In this paper, the construction of MA blocked factorial designs is studied in a row-wise fashion, and sufficient conditions are given for constructing MA blocked designs from unblocked MA designs.
Abstract: This paper considers the construction of minimum aberra- tion (MA) blocked factorial designs. Based on coding theory, the con- cept of minimum moment aberration due to Xu (2003) for unblocked designs is extended to blocked designs. The coding theory approach studies designs in a row-wise fashion and therefore links blocked designs with nonregular and supersaturated designs. A lower bound on blocked wordlength pattern is established. It is shown that a blocked design has MA if it originates from an unblocked MA design and achieves the lower bound. It is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros. Sufficient conditions are given for constructing MA blocked designs from unblocked MA designs. The theory is then applied to construct MA blocked designs for all 32 runs, 64 runs up to 32 factors, and all 81 runs with respect to four combined wordlength patterns.
2 citations
Cites background from "Minimum Aberration 2 k–p Designs"
...It is shown that a blocked design has MA if it originates from an unblocked MA design and achieves the lower bound....
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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