Complex linear models and cyclic designs
Abelian group theory has been used to construct many orthogonal fractional replicates or block designs in confounding. Usually, abelian group structures are given to the set of experimental units, the set of treatments, and, for block designs, the set of blocks. The mapping assigning a treatment, or a block, to each experimental unit is then chosen among affine mappings, which are the composition of a group morphism and a translation. In this energetic paper, several concepts of designs and properties are discussed systematically. From the abstract: ``Many orthogonal factorial designs can be defined by abelian group morphisms. By juxtaposition of such designs, useful nonorthogonal designs can also be obtained, including the classical generalized cyclic designs, as well as a new kind of replicate factorial block designs. Their efficiencies are easily computed by means of a complex reparametrization based on the irreducible characters of the groups involved. The theory extends to the ``group generated designs'' defined by \textit{R. A. Bailey} and \textit{C. A. Rowley} [ibid., 183-225 (1990; Zbl 0725.62067)] where the group is not necessarily abelian. In some cases, we give explicit formulas for the efficiencies of these latter designs.''
scientific article; zbMATH DE number 4213270
scientific article