Some families of group divisible designs

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).     

Families of regular designs based on geometries

Recently Bush and Ostrom (1979) settled most of the open questions with respect to inequivalent solutions of a class of semiregular (SR) designs which can be constructed from nets. This paper is a study of the same nature for two families of regular (R) designs derived from finite projective planes. One family presents no problems, but the other which is a ‘double’ family with two parameters is much more difficult. In fact it is here solved only for designs based on planes of orders 3, 4, 5 and 8. Certain general methods exist which are indicated, but we were unable to resolve even the case 7 using this technique.Basically we show the existence of either inequivalent solutions or show there is but one solution settling a number of open cases. In particular for the case λ₁ = 2, λ₂ = 1 we give new solutions to a number of D(2) designs or group divisible designs with two associate classes which have no repeated blocks in contrast with the published solutions which have this undesirable property for a number of applications.     

Regular group divisible designs and Bhaskar Rao designs with block size three

Some recursive constructions are given for Bhaskar Rao designs. Using examples of these designs found by Shyam J. Singh, Rakesh Vyas and new ones given here we show the necessary conditions λ≡0 (mod 2), λυ(υ?1)≡0 (mod 24) are sufficient for the existence of Bhaskar Rao designs with one association class and block size 3. This result is used with a result of Street and Rodger to obtain regular partially balanced block designs with 2υ treatments, block size 3, λ₁=0, group size 2 and υ groups.     

A construction of group divisible designs

Given any affine design with parameters v, b, r, k, λ and μ = k²/v and any design with parameters v′, b′, r′, k′, λ′ where r′ = tr for some natural number `t and k′?r, we construct a group divisible design with parameters v′' = vv′, m = v′, n = v, b′' = vb′, k′' = kk′, r′'= kr′, λ₁ = tkλ and λ₂ = μλ′. This is applied to some series of designs. As a lemma, we also show that any 0-1-matrix with row sums tr and column sums ?r may be written as the sum of r 0-1-matrices with row sums t and column sums ?1.     

Recursive constructions for cyclic block designs

Simple recursive constructions for cyclic block designs are given. These yield many new infinite families of cyclic Steiner 2-designs.     

On construction of constant block-sum partially balanced incomplete block designs

Abstract

Constant block-sum designs are of interest in repeated measures experimentation where the treatments levels are quantitative and it is desired that at the end of the experiments, all units have been exposed to the same constant cumulative dose. It has been earlier shown that the constant block-sum balanced incomplete block designs do not exist. As the next choice, we, in this article, explore and construct several constant block-sum partially balanced incomplete block designs. A natural choice is to first explore these designs via magic squares and Parshvanath yantram is found to be especially useful in generating designs for block size 4. Using other techniques such as pair-sums and, circular and radial arrangements, we generate a large number of constant block-sum partially balanced incomplete block designs. Their relationship with mixture designs is explored. Finally, we explore the optimization issues when constant block-sum may not be possible for the class of designs with a given set of parameters.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

Generalized Bhaskar Rao designs

Generalized Bhaskar Rao designs with non-zero elements from an abelian group G are constructed. In particular this paper shows that the necessary conditions are sufficient for the existence of generalized Bhaskar Rao designs with k=3 for the following groups: ?G? is odd, G=Z^r₂, and G=Z^r₂×H where 3? ?H? and r?1. It also constructs generalized Bhaskar Rao designs with υ=k, which is equivalent to υ rows of a generalized Hadamard matrix of order n where υ?n.     

Some sufficient conditions for the type 1 optimality of block designs

In this paper, we investigate the problem of determining block designs which are optimal under type 1 optimality criteria within various classes of designs having υ treatments arranged in b blocks of size k. The solutions to two optimization problems are given which are related to a general result obtained by Cheng (1978) and which are useful in this investigation. As one application of the solutions obtained, the definition of a regular graph design given in Mitchell and John (1977) is extended to that of a semi-regular graph design and some sufficient conditions are derived for the existence of a semi-regular graph design which is optimal under a given type 1 criterion. A result is also given which shows how the sufficient conditions derived can be used to establish the optimality under a specific type 1 criterion of some particular types of semi- regular graph designs having both equal and unequal numbers of replicates. Finally,some sufficient conditions are obtained for the dual of an A- or D-optimal design to be A- or D-optimal within an appropriate class of dual designs.     

D-optimal block designs with at most six varieties

For given positive integers v, b, and k (all of them ≥2) a block design is a k × b array of the variety labels 1,…,v with blocks as columns. For the usual one-way heterogeneity model in standard form the problem is studied of finding a D-optimal block design for estimating the variety contrasts, when no balanced block design (BBD) exists. The paper presents solutions to this problem for v≤6. The results on D-optimality are derived from a graph-theoretic context. Block designs can be considered as multigraphs, and a block design is D-optimal iff its multigraph has greatest complexity (=number of spanning trees).     

Symmetric designs of types V and VI

In Butler (1984a) a semi-translation block was defined and a classification given of all symmetric 2-(υ,k,λ) designs with λ>1, which contain more than one such block. In this paper we consider symmetric designs of type V and VI. We show that symmetric designs of type V are also of type VI, and in addition we show that all such designs can be obtained from a P_n,q by a construction which we give. Finally examples of proper symmetric designs of type V which are not of type VI are given.     

Some new group divisible designs

The paper lists fourteen new group divisible PBIB/2 designs, which were obtained using the computer program described in John (1976).     

Families of Hadamard group divisible designs

By a family of designs we mean a set of designs whose parameters can be represented as functions of an auxiliary variable t where the design will exist for infinitely many values of t. The best known family is probably the family of finite projective planes with υ = b = t² + t + 1, r = k = t + 1, and λ = 1. In some instances, notably coding theory, the existence of families is essential to provide the degree of precision required which can well vary from one coding problem to another. A natural vehicle for developing binary codes is the class of Hadamard matrices. Bush (1977) introduced the idea of constructing semi-regular designs using Hadamard matrices whereas the present study is concerned mostly with construction of regular designs using Hadamard matrices. While codes constructed from these designs are not optimal in the usual sense, it is possible that they may still have substantial value since, with different values of λ₁ and λ₂, there are different error correcting capabilities.     

Construction of some classes of balanced cross-over designs from terraces

5 and 6 have recently introduced power-sequence terraces. In this paper we have used these terraces for the construction of some new families of balanced cross-over designs of first and second order which are variance-balanced. We have also used them for the construction of some new families of balanced ternary cross-over designs.     

Further properties of generalized residual designs

The concept of ‘residuation’ is extended so that all ‘generalized residual designs’ (in the sense of Shrinkhande and Singhi) are in fact ‘residual’ with respect to the extended type of residuation. A measure of departure from the usual type of residuation is given in general, and stronger estimates of this measure are given for affine designs.     

Fractional Factorial Designs for the Detection of Interactions between Design and Noise Factors

In industrial experiments on both design (control) factors and noise factors aimed at improving the quality of manufactured products, designs are needed which afford independent estimation of all design×noise interactions in as few runs as possible, while allowing aliasing between those factorial effects of less interest. An algorithm for generating orthogonal fractional factorial designs of this type is described for factors at two levels. The generated designs are appropriate for experimenting on individual factors or for experimentation involving group screening of factors.     

On the equivalence of definitions for regular fractions of mixed-level factorial designs

The notion of regularity for fractional factorial designs was originally defined only for two-level factorial designs. Recently, rather different definitions for regular fractions of mixed-level factorial designs have been proposed by Collombier [1996. Plans d’Expérience Factoriels. Springer, Berlin], Wu and Hamada [2000. Experiments. Wiley, New York] and Pistone and Rogantin [2008. Indicator function and complex coding for mixed fractional factorial designs. J. Statist. Plann. Inference 138, 787–802]. In this paper we prove that, surprisingly, these definitions are equivalent. The proof of equivalence relies heavily on the character theory of finite Abelian groups. The group-theoretic framework provides a unified approach to deal with mixed-level factorial designs and treat symmetric factorial designs as a special case. We show how within this framework each regular fraction is uniquely characterized by a defining relation as for two-level factorial designs. The framework also allows us to extend the result that every regular fraction is an orthogonal array of a strength that is related to its resolution, as stated in Dey and Mukerjee [1999. Fractional Factorial Plans. Wiley, New York] to mixed-level factorial designs.     

Construction of (M,S)-optimal design for block size 3

(M,S)-optimal designs are constructed for block size three when the number of treatments is of the form 6t + 3.     

Block designs with block size two

Algorithms are given for the construction of binary block designs with replications and concurrences differing by at most one. The designs are resolvable and/or connected wherever the parameters permit.