Optimality and Construction of Pseudo-Youden Designs

Abstract

In the two-way heterogeneity setting, the optimality of a generalized Youden design (GYD) has been proved by Kiefer (1975a). A GYD is a design which is a balanced block design (BBD) when each of {rows} and {columns} is considered as blocks. It is observed in the present paper that when the number of rows is equal to the number of columns, a design is optimal as long as the rows and columns together form a BBD. Such a design is termed a pseudo-Youden design (PYD). A square GYD is also a PYD, but the converse is not true. Thus, the stringent conditions imposed on the definition of a GYD are substantially relaxed. A PYD is easier to construct and has the same efficiency as a GYD if they exist simultaneously. Patchwork and geometric methods are combined to construct a family of PYD's. It is also indicated when the construction of a GYD is impossible. A $6 \times 6$ PYD with 9 varieties is constructed. This design has the property that the number of rows is less than the number of varieties, which is never achieved by a square GYD. There is also an analogous theory for higher-dimensional designs.