# A Unified Approach to Factorial Designs with Randomization Restrictions

TL;DR: In this article, 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).

Abstract: AbtsrcatFactorial 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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### "A Unified Approach to Factorial Des..." refers background or methods in this paper

...To ensure the significance assessment of all effects in an experiment, Ranjan et al. (2009) recommended that the RDCSSs be disjoint and large enough to construct useful half-normal plots. Schoen (1999) suggests maintaining a minimum of six - seven points per half-normal plot for meaningful analysis. As a result, the existence and construction of the desired design is equivalent to the availability of a set of disjoint large flats of Pn. Ranjan et al. (2009) used spreads of Pn, set of flats that partition Pn, for such designs....

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..., block designs, split-plot designs, strip-plot designs, split-lot designs, and combinations thereof) are often used for designing industrial experiments when complete randomization of the trials is impractical (Miller 1997; Mee and Bates 1998; Vivacqua and Bisgaard 2004; Bingham et al. 2008)....

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...See Ranjan et al. (2009) for more results on the existence of balanced/mixed full/partial spreads of Pn....

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...Example 2.4 In a 25 strip-plot design suppose the row configurations are defined by A, B, and the column configurations by C, D, E (e.g., washerdryer example in Miller 1997)....

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...…(e.g., block designs, split-plot designs, strip-plot designs, split-lot designs, and combinations thereof) are often used for designing industrial experiments when complete randomization of the trials is impractical (Miller 1997; Mee and Bates 1998; Vivacqua and Bisgaard 2004; Bingham et al. 2008)....

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

...The relabeling of factorial effects or points of Pn can be done using a collineation (Coxeter 1974; Batten 1997)....

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