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Journal ArticleDOI

A Unified Approach to Factorial Designs with Randomization Restrictions

01 Mar 2013-Calcutta Statistical Association Bulletin (SAGE Publications)-Vol. 65, pp 43-62

AbstractAbtsrcatFactorial designs are commonly used to assess the impact of factors and factor combinations in industrial and agricultural experiments. Though preferred, complete randomization of trials is often infeasible, and randomization restrictions are imposed. In this paper, we discuss a finite projective geometric (PG) approach to unify the existence, construction and analysis of multistage factorial designs with randomization restrictions using randomization defining contrast subspaces (or flats of a PG). Our main focus will be on the construction of such designs, and developing a word length pattern scheme that can be used for generalizing the traditional design rank- ing criteria for factorial designs. We also present a novel isomorphism check algorithm for these designs.

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24 Jan 1980
Abstract: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. First properties of the plane 8. Ovals 9. Arithmetic of arcs of degree two 10. Arcs in ovals 11. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes Appendix Notation References

1,547 citations


"A Unified Approach to Factorial Des..." refers background in this paper

  • ...This cyclic construction is ensured by the condition t|n (Hirschfeld 1998)....

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Journal ArticleDOI
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.

415 citations


"A Unified Approach to Factorial Des..." refers background in this paper

  • ...…thus far for ranking regular and non-regular fractional factorial designs, for instance, maximum resolution criterion (Box and Hunter 1961), minimum aberration (Fries and Hunter 1980), maximum number of clear effects (Wu and Chen 1992) and generalized minimum aberration (Deng and Tang 1999)....

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Journal ArticleDOI

198 citations


"A Unified Approach to Factorial Des..." refers background in this paper

  • ...…criteria have been proposed thus far for ranking regular and non-regular fractional factorial designs, for instance, maximum resolution criterion (Box and Hunter 1961), minimum aberration (Fries and Hunter 1980), maximum number of clear effects (Wu and Chen 1992) and generalized minimum…...

    [...]

01 Jan 1999
Abstract: Resolution has been the most widely used criterion for comparing regular fractional factorials since it was introduced in 1961 by Box and Hunter. In this pa- per, we examine how a generalized resolution criterion can be defined and used for assessing nonregular fractional factorials, notably Plackett-Burman designs. Our generalization is intended to capture projection properties, complementing that of Webb (1964) whose concept of resolution concerns the estimability of lower order ef- fects under the assumption that higher order effects are negligible. Our generalized resolution provides a fruitful criterion for ranking different designs while Webb's resolution is mainly useful as a classification rule. An additional advantage of our approach is that the idea leads to a natural generalization of minimum aberration. Examples are given to illustrate the usefulness of the new criteria.

186 citations


"A Unified Approach to Factorial Des..." refers background in this paper

  • ...…thus far for ranking regular and non-regular fractional factorial designs, for instance, maximum resolution criterion (Box and Hunter 1961), minimum aberration (Fries and Hunter 1980), maximum number of clear effects (Wu and Chen 1992) and generalized minimum aberration (Deng and Tang 1999)....

    [...]

  • ...Several design criteria have been proposed thus far for ranking regular and non-regular fractional factorial designs, for instance, maximum resolution criterion (Box and Hunter 1961), minimum aberration (Fries and Hunter 1980), maximum number of clear effects (Wu and Chen 1992) and generalized minimum aberration (Deng and Tang 1999)....

    [...]

Journal ArticleDOI
Abstract: In planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects. In this article, we propose a graph-aided method to solve this problem for two-level experiments. First, we choose the defining relations for a 2 n–k design according to a goodness criterion such as the minimum aberration criterion. Then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two-factor interactions for the given defining relations. In each graph a vertex represents a factor and an edge represents the interaction between the two factors. For the experiment planner, the job is simple: Draw a graph representing the specified interactions and compare it with the list of graphs obtained previously. Our approach is a substantial improvement over Taguchi's linear graphs.

172 citations


"A Unified Approach to Factorial Des..." refers background in this paper

  • ...…thus far for ranking regular and non-regular fractional factorial designs, for instance, maximum resolution criterion (Box and Hunter 1961), minimum aberration (Fries and Hunter 1980), maximum number of clear effects (Wu and Chen 1992) and generalized minimum aberration (Deng and Tang 1999)....

    [...]

  • ...Several design criteria have been proposed thus far for ranking regular and non-regular fractional factorial designs, for instance, maximum resolution criterion (Box and Hunter 1961), minimum aberration (Fries and Hunter 1980), maximum number of clear effects (Wu and Chen 1992) and generalized minimum aberration (Deng and Tang 1999)....

    [...]