A Calculus for Factorial Arrangements

Factorial designs were introduced and popularized by Fisher (1935). Among the early authors, Yates (1937) considered both symmetric and asymmetric factorial designs. Bose and Kishen (1940) and Bose (1947) developed a mathematical theory for symmetric priIi't&-powered factorials while Nair and Roo (1941, 1942, 1948) introduced and explored balanced confounded designs for the asymmetric case. Since then, over the last four decades, there has been a rapid growth of research in factorial designs and a considerable interest is still continuing. Kurkjian and Zelen (1962, 1963) introduced a tensor calculus for factorial arrangements which, as pointed out by Federer (1980), represents a powerful statistical analytic tool in the context of factorial designs. Kurkjian and Zelen (1963) gave the analysis of block designs using the calculus and Zelen and Federer (1964) applied it to the analysis of designs with two-way elimination of heterogeneity. Zelen and Federer (1965) used the calculus for the analysis of designs having several classifications with unequal replications, no empty cells and with all the interactions present. Federer and Zelen (1966) considered applications of the calculus for factorial experiments when the treatments are not all equally replicated, and Paik and Federer (1974) provided extensions to when some of the treatment combinations are not included in the experiment. The calculus, which involves the use of Kronecker products of matrices, is extremely helpful in deriving characterizations, in a compact form, for various important features like balance and orthogonality in a general multifactor setting.

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1. Introduction.- 2. A Calculus For Factorial Arrangements.- Elements and operations of the calculus.- Orthogonal factorial structure and balance.- 3. Characterizations For Balance With Orthogonal Factorial Structure.- Algebraic characterizations.- A combinatorial characterization.- A review of construction procedures.- Concluding remarks.- 4. Characterizations For Orthogonal Factorial Structure.- and preliminaries.- Algebraic characterizations: the connected case.- Algebraic characterizations: the disconnected case.- Partial orthogonal factorial structure.- Efficiency consistency.- 5. Constructions I: Factorial Experiments In Cyclic And Generalized Cyclic Designs.- Factorial experiments in cyclic designs.- Generalized cyclic designs.- Single replicate factorials in GC/n designs.- Further results.- Row-column designs.- Designs with partial orthogonal factorial structure.- 6. Constructions II: Designs Based On Kronecker Type Products.- Designs through ordinary Kronecker product.- Componentwise Kronecker product of order q.- Khatri-Rao product of order q.- Non-equireplicate designs.- Designs for multiway heterogeneity elimination.- 7. More On Single Replicate Factorial Designs.- General classical designs.- Bilinear classical designs.- Concluding remarks.- 8. Further Developments.- Deletion designs.- Merging of treatments.- Results on efficiency and admissibility.- Concluding remarks.- 113.- 124.