A series of group divisible designs

Bose and Shrikhande C19763 proved that if D(m, k, ?) is a Baer subdesign of another SBIBD D₁ (v₁, k₁ ?), k₁>k, then it also contains a complementary subdesign D^* which is symmetric GDD, D^* (v^*, k^*; ?-1, ?; m, n). Utilising this, we give a necessary condition for a SBIBD D to be a Baer subdesign of D₁ and also give the parameters. Some GD designs are constructed.     

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.     

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).     

Generalizing Clatworthy group divisible designs

In this paper a neat construction is provided for three new families of group divisible designs that generalize some designs from Clatworthy's table of the only 11 designs with two associate classes that have block size four, three groups, and replication numbers at most 10. In each case (namely, _λ1=4${\lambda }_1=4$ and _λ2=5${\lambda }_2=5$, _λ1=4${\lambda }_1=4$ and _λ2=2${\lambda }_2=2$, and _λ1=8${\lambda }_1=8$ and _λ2=4${\lambda }_2=4$), we have proved that the necessary conditions found are also sufficient for the existence of such GDD's with block size four and three groups, with one possible exception.     

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.     

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.     

On group divisible response surface (GDRS) designs

In this paper some experimental situations are identified corresponding to which suitable response surface designs do not exist. A class of response surface designs is introduced to cope with these situations. Their analysis with and without blocking and methods of construction is discussed.     

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).     

Some new group divisible designs

The paper lists four new group divisible designs which are obtained by trial and error. These designs are believed to be new, since they are not listed in Clatworthy (1973), Freeman (1976) or John and Turner (1977).     

Super-simple group divisible designs with block size 4 and index 9

A design is said to be super-simple if the intersection of any two blocks has at most two elements. In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple GDDs are useful in constructing super-simple BIBDs. The existence of super-simple (4,λ)‐GDDs has been determined for λ=2-6. In this paper, we investigate the existence of a super-simple (4,9)-GDD of group type g^u and show that such a design exists if and only if u≥4, g(u−2)≥18 and u(u−1)g²≡0 (mod 4).     

Designs for symmetrical parallel line assays obtainable through group divisible designs

A wide class of block designs for symmetrical parallel line assays with even number of doses, obtainable through group divisible designs is considered. Several new designs can thus be obtained using group divisibledesigns. Group divisible designs are also shown to provide a unification of many exisiting designs which follow as special cases.     

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.     

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.     

Randomized response multivariate designs for categorical data

Three approaches to multivariate estimation for categorical data using randomized response (RR) are described. In the first approach, practical only for 2×2 contingency tables, a multi-proportions design is used. In the second approach, a separate RR trial is used for each variate and it is noted that the multi­variate design matrix of conditional probabilities is given by the Kroneeker product of the univariate design matrices of each trial, provided that the trials are independent of each other in a certain sense. The third approach requires only a single randomization and thus may be viewed as the use of vector response. Finally, a special-purpose bivariate design is presented.     

Minimal point second order designs

This paper deals with the problem of finding nearly D-optimal designs for multivariate quadratic regression on a cube which take as few observations as possible and still allow estimation of all parameters. It is shown that among the class of all such designs taking as many observations as possible on the corners of the cube there is one which is asymptotically efficient as the dimension of the cube increases. Methods for constructing designs in this class, using balanced arrays, are given. It is shown that the designs so constructed for dimensions ≤6 compare well with existing computer generated designs, and in dimensions 5 and 6 are better than those in literature prior to 1978.     

Step-wise group screening designs with errors in observations

The performance of step-wise group screening In terms of the expected number of runs and the expected number of incorrect decisions, Is considered. A method for obtaining optimal step-wise designs Is presented, for the cases in which the direction of each defective factor is assumed to be known a-priori and the observations are subject to error.     

Hypercubic designs and applications

In this paper we develop relatively easy methods for constructing hypercubic designs from symmetrical factorial experiments for t=v ^m treatments with v=2, 3. The proposed methods are easy to use and are flexible in terms of choice of possible block sizes.