Three methods of constructing pbib designs

Three construction methods of two- or three-associate partially balanced incomplete block (PBIB) designs are presented.     

New combinatorial designs and their applications to group testing

A class of designs with property C(t) are introduced for the first time, and their applications in group testing of samples are studied.     

A new special class of peb designs

A new class of partially efficiency-balanced designs is introduced from a practical point of view. This new design includes all equireplicated incomplete block designs available in literature as special cases. The fundamental properties of the design are clarified with relation to other block designs.     

On the construction of group divisible incomplete block designs

In this paper a method of constructing group-divisible incomplete block designs has been suggested. A series of balanced incomplete block designs has also been obtained.     

Generating generalized cyclic designs with factorial balance

An algorithm for generating paired comparison factorially balanced generalized cyclic designs is described. The algorithm is based upon the 2 ⁿ ? 1 class association scheme defined by Shah (1960) for n-factor experiments. The algorithm is highly successful in achieving its objective. Firstorder designs with block size greater than two can also be obtained using the algorithm.     

General upper bound for the number of blocks having a given number of treatments common with a given block

The purpose of this paper is systematically to derive the general upper bound for the number of blocks having a given number of treatments common with a given block of certain incomplete block designs. The approach adopted here is based on the spectral decomposition of N′N for the incidence matrix N of a design, where N' is the transpose of the matrix N. This approach will lead us to upper bounds for incomplete block designs, in particular for a large number of partially balanced incomplete block (PBIB) designs, which are not covered with the standard approach (Shah 1964, 1966), Kapadia (1966)) of using well known relations between blocks of the designs and their association schemes. Several results concerning block structure of block designs are also derived from the main theorem. Finally, further generalizations of the main theorem are discussed with some illustrations.     

Construction of incomplete Sudoku square and partially balanced incomplete block designs

Abstract

The present article deals with the study of association among the elements of a Sudoku square. In this direction, we have defined an association scheme and constructed incomplete Sudoku square designs which are capable of studying four explanatory variables and also happen to be the designs for two-way elimination of heterogeneity. Some series of Partially Balanced Incomplete Block (PBIB) designs have also been obtained.     

Connectedness of two-associate PBIB designs

Necessary and sufficient conditions for two-associate partially balanced incomplete block designs to be connected are sysmematically discussed in terms of parameters of the original designs, together with the result of Mohan (1981).     

On the construction of m-associate PBIB designs I

The m-associate triangular association scheme has been discussed, and several series of partially balanced incomplete block (PBIB) designs with m-associate triangular association scheme have been obtained in Section 1. In Section 2, an m-associate triangular-group divisible association scheme $\text{(T}{}_{\mathrm{q}}\text{—}\text{GD}\text{m}\text{, 1<q<m)}$ has been introduced and several series of PBIB designs with m-associate triangular group divisible association scheme, from m-associate triangular PBIB designs have been constructed. Some numerical values to the three associate triangular designs, and three associate triangular group divisible designs in the range b, v≦100; r, k≦10, with their average efficiencies are given, respectively, in Tables I and II, in Section 3, where as usual v denotes the number of treatments, b the number of blocks, r the number of replications of each treatment, and k the block size.     

BALANCED ARRAYS AND WEIGHING DESIGNS

Dey (19711, Saha (1975), Kageyama & Saha (1983) and others have shown how optimum chemical balance weighing designs can be constructed from the incidence matrices of balanced incomplete block (BIB) designs. In this paper, it is shown that weighing designs can be constructed from some suitably chosen two-symbol balanced arrays of strength two, which need not always be incidence matrices of BIB designs. The findings lead us to construct new optimum chemical balance weighing designs from incidence matrices of BIB designs.     

The analysis of mixed and random effects models for equireplicate variance-balanced block designs

Equirephcate variance-balanced block designs possess many desirable properties and include several important subclasses of designs, such as balanced incomplete block designs. However, while attention has been paid to the derivation of properties and the construction of these designs, methods of analyzing experiments using these designs have not been developed completely. Thus, we consider models appropriate to these designs, construct test statistics for basic hypotheses and derive the distributions for these statistics. Four additive models that include all of the possible combinations of fixed and random effects are examined. The analyses are generalizations of the analyses for balanced incomplete block designs, and include the intra-block, intra-treatment and inter-block analyses, Estimators of fixed treatment effects and their distributions also are given.     

On a class of variance balanced incomplete block designs

In this paper variance balanced incomplete block designs have been constructed for situations when suitable BIB designs do not exist for a given number of treatments, because of the contraints bk=vr, λ(v-1) = r(k-l). These variance balanced designs are in unequal block sizes and unequal replications.     

On the construction of hypercubic design

Hypercubic design was introduced by Shah (1958) and Kusumoto(1965) in which the t=v^m treatments are represented by n-plets (x₁, x₂, ..., x_m),where 1≤ x₁, x₂, ..., x_m ≤v, and two treatments are said to be i-th associates if they differ in exactly i components. This paper deals with the construction of some hypercubic designs and gives their application to confounding plans for symmetrical factorial experiments. The proposed methods prove to be quite flexible in terms of choice of possible block sizes and are easy to use.     

A note on the construction of some lattices for the hypercubic association scheme

This paper presents a method of construction for some PBIB lattices for the hypercubic association scheme.     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

A characterization of a universally optimal design within a class of block designs

It is shown that within the class of connected binary designs with arbitrary block sizes and arbitrary replications only a symmetic balanced incomplete block design produces a completely symmetric information matrix for the treatment effects whenever the number of blocks is equal to the number of treatments and the number of experimental units is an integer multiple of the number of treatments. Such a design is known to be universally optimal.     

A note on generalized binary proper efficiency-balanced block designs

Some methods of construction of generalized binary proper efficiency-balanced block (GB-EB) designs with two different replications are derived and the solution of the unknown designs given by Das and Kageyama (1991) are obtained.     

Connectedness of PBIB designs

A necessary and sufficient condition for the connectedness of an m-associate-class PBIB design is derived in a form, and using an approach, different from those in Ogawa, Ikeda, and Kageyama (1984) and Saha and Kageyama (1984). The applicability of this condition is shown in the context of group-divisible m-associate-class PBIB designs.     

Remark on Construction of Efficiency-Balanced Designs

In this note, all the efficiency-balanced block designs constructed by utilizing two methods of Dey & Singh (1980) are completely presented within a practical range of parameters.     

Methods of constructing group divisible designs with k = 3 and 4

The paper provides methods of construction of group divisible designs with block sizes three and four through balanced incomplete block and partially balanced incomplete block designs of block sizes 3 and 4. Furthermore, four non-isomorphic solutions are given.