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, the degrees of freedom criterion is introduced to guarantee a good choice of the fractional design also in terms of the possibility to perform parametric tests with a good power and to obtain confidence intervals for the effects of interest.
Abstract: In many experimental situations, when a two-level factorial design is performed, the effects of higher order interactions are usually less important than the effects of lower order interactions (and effects of the same order are equally important). Hence, we should search for good fractional factorial designs withN observations allowing for the estimation of the effects of main factors and of some interactions under study. Obviously, a design is more preferable it if allows for several de-aliased estimates of the effects of interest. A goodness criterion of a design is its resolution (Box & Hunter (1961)). Since there is more than one design with the same resolution, to select the «best» subset of designs which have the same resolution (usually the maximum resolution), we may use the criterion of minimum aberration (Fries & Hunter (1980)). In experiments involving control and noise factors (Taguchi (1987)) neither the resolution nor the aberration criterion can guarantee a good statistical design. Here we introduce the degrees of freedom criterion in order to guarantee a good choice of the fractional design also in terms of the possibility to perform parametric tests with a good power and to obtain confidence intervals for the effects’ estimates. The two level orthogonal factorial designs are suitable for experiments with linear response, since they give the possibility to obtain uniformly optimal de-aliased estimates of the effects of interest. Here we consider the case where three and higher factor interactions are negligible and orthogonal designs of resolution IV and V are constructed; the necessary conditions for the existence of such designs areN ∈0 mod 8, andN ∈0 mod 16,N ≥16, respectively. Some cases whenN ∈0 mod 8,N ≥ 8, orN ∈0 mod 16,N ≥ 16, are studied: at first we construct orthogonal designs of resolution IV and V for a given number of factors k, and then for a given number of runsN (in particularN=48, 80, 96, 112).
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TL;DR: Blocking is commonly used in experimental design to eliminate unwanted variation by creating more homogeneous conditions for experimental treatments within each block.
Abstract: Blocking is commonly used in experimental design to eliminate unwanted variation by creating more homogeneous conditions for experimental treatments within each block. While it has been a standard ...
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TL;DR: In this paper, a maximum estimability (maxest) criterion is proposed for design classification and selection, which is an extension and refinement of Webb's resolution criterion for general factorial designs.
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Dissertation•
01 Jan 2008
TL;DR: Thesis (Ph. D.)--Massachusetts Institute of Technology, Engineering Systems Division, 2008.
Abstract: Thesis (Ph. D.)--Massachusetts Institute of Technology, Engineering Systems Division, 2008.
3 citations
Cites background or methods from "Minimum Aberration 2 k–p Designs"
...The fractional factorial design is selected in accordance to maximum resolution (Box and Hunter; 1961) and minimum aberration (Fries and Hunter; 1980) criteria. The arrays that meet these criteria are summarized in Wu and Hamada (2000). It is seen that four arrays for the size 27-2 and two arrays for size 2(7-1) meet these requirements of optimality....
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...The selection of the appropriate fraction is based on mathematical criteria such as maximum resolution (Box and Hunter, 1961) or minimum aberration (Fries and Hunter, 1980)....
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TL;DR: In this paper, the authors investigated the construction and theoretical properties of follow-up experiments obtained via the addition of two $$n/2$$ -run semifoldover fractions, which provides a means of estimating more effects than can be achieved with a foldover design.
Abstract: The addition of another fraction to an initial experiment is often necessary to resolve ambiguities involving aliasing of factorial effects. One of the most widely used techniques for the selection of a follow-up experiment is foldover. However, semifoldover (i.e., adding half of a foldover fraction) frequently permits estimation of as many effects of interest as provided by a foldover. Thus, as an alternative to foldover, this article investigates the construction and theoretical properties of follow-up experiments obtained via the addition of two $$n/2$$
-run semifoldover fractions. The strategy (termed double semifolding) provides a means of estimating more effects than can be achieved with a foldover design. Through the use of indicator functions, general properties of double semifoldover designs will be developed. Optimal double semifoldover plans, based on several established design criteria, will be discussed and tabulated for practical use.
3 citations
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