Sanpei Kageyama

A Survey of Resolvable Solutions of Balanced Incomplete Block Designs

In a balanced incomplete block (BIB) design with parameters v, b, r, k and 1, if the blocks can be separated into r sets of n blocks each (b = nr) such that each set of n blocks forms a complete replication, the design is called resolvable. Moreover, if two blocks belonging to different sets have the same number of treatments in common, the design is called affine resolvable. Bose [2] proved that if a resolvable BIB design with parameters v, b, r, k and 2 exists, then b ? v + r1 and that if for a resolvable BIB design the condition b = v+r-1 holds, then the design is affine resolvable and further, the number of treatments common to any two blocks of different sets is k2/v, so that k2 must be divisible by v. That is to say, necessary conditions for the existence of a resolvable BIB design are that

On properties of efficiency-balanced designs

We consider a class of efficiency-balanced block designs which are design patterns for the analytical simplification of statistical analysis. This paper consists of eight sections which investigate various problems (such as, bounds on parameters, characterizations, existence and nonexistence, and dual designs) for efficiency-balanced block designs.

THE FAMILY OF t-DESIGNS* - PART II

Abstract The family of t -designs is, without any doubt, the most important family of statistical designs. Their importance is due to their statistical optimalities, desirable symmetries for analyses and interpretations, and uses for constructing other important designs and structures such as Youden designs, generalized Youden designs, optimal fractional factorial designs, error defecting and correcting binary codes, balanced arrays, combinatorial filing systems, Hadamard matrices, finite projective and affine planes, strongly regular graphs, and so on. Research in the area of t -designs has been steadily and rapidly growing, especially during the last three decades. The number of publications in this area is in the several hundreds. Since papers on t -designs are published in a variety of journals, and because of the extensive role of these designs in design of experiments and other areas we believe it is imperative to gather these results and present them in varied form to suit diverse interests. This paper is an instance of such an attempt.

Single replicate orthogonal block designs for circulant partial diallel crosses

Some incomplete block designs for partial diallel crosses have been given in the literature. These designs are obtained by regarding the number of crosses as treatments, and consequently require several replications of each cross. The need for resorting to a partial diallel cross itself implies that it is desired to have fewer crosses. A method for constructing single replicate incomplete block designs for circulant partial diallel crosses is provided in this paper. The designs are orthogonal, and thus they retain full efficiency for estimation of the contrasts of interest.

Some bounds on balanced block designs

Abstract Bounds on the latest root of the C -matrix and the number of blocks for a variance-balanced block design are given. These results contain the well-known results as special cases.

Robustness of BIB and extended BIB designs against the unavailability of any number of observations in a block

Abstract Robustness of block designs is investigated, when any observations in a block are lost, in terms of efficiency of the residual design. The designs are balanced incomplete block (BIB) and extended BIB designs. The investigation shows that the covered designs are fairly robust against the unavailability of any number of observations in a block. Furthermore, the robustness of any Youden design and Latin square design against the loss of a whole column is shown.

Some families of group divisible designs

Abstract Generalizing methods of constructions of Hadamard group divisible designs due to Bush (1979), some new families of semi-regular or regular group divisible designs are produced. Furthermore, new nonisomorphic solutions for some known group divisible designs are given, together with useful group divisible designs not listed in Clatworthy (1973).

Robustness group divisible designs

Robustness of group divisible (GD) designs is investigated, when one block is lost, in terms of efficiency of the residual design. The exact evaluation of the efficiency can be made for singular GD and semi-regular GD designs as ell as regular GD designs with λ1 = 0. In a regular GD design with λ1 > 0, the efficiency may depend upon the lost block and sharp upper and lower bounds on the efficiency are presented. The investigation shows that GD designs are fairly robust in terms of efficiency. As a special case, we can also show the robustness of balanced incomplete block design when one block is lost.

Constructions of efficiency-balanced designs

We present a number of methods of constructing efficiency-balanced binary block designs which are design patterns for simplification of statistical analysis. Furthermore, a method of construction of an efficiency-balanced block design with v+1 treatments from one with v treatments is generally characterized.

A Construction for Resolvable Designs and Its Generalizations

Abstract. In [14], D.K. Ray-Chaudhuri and R.M. Wilson developed a construction for resolvable designs, making use of free difference families in finite fields, to prove the asymptotic existence of resolvable designs with index unity. In this paper, generalizations of this construction are discussed. First, these generalizations, some of which require free difference families over rings in which there are some units such that their differences are still units, are used to construct frames, resolvable designs and resolvable (modified) group divisible designs with index not less than one. Secondly, this construction method is applied to resolvable perfect Mendelsohn designs. Thirdly, cardinalities of such sets of units are investigated. Finally, composition theorems for free difference families via difference matrices are described. They can be utilized to produce some new examples of resolvable designs.