Optimality aspects of 3-concurrence most balanced designs

Takcuchi (1961,1963) established E-optimality of Group Divisible Designs (GDDs) with λ₂=λ₁+1. Much later, Cheng (1980) and Jacroux (1980,1983) demonstrated E-optimality property of the GDDs with n=2,λ₁=λ₂+1 or with m=2,λ₂=λ₁+2. The purpose of this paper is to provide a unified approach for identifying certain classes of designs as E-optimal. In the process, we come up with a complete characterization of all E-optimal designs attaining a specific bound for the smallest non-zero eigenvalue of the underlying C-matrices. This establishes E-optimality of a class of 3-concurrence most balanced designs with suitable intra- and inter-group balancing. We also discuss the MV-optimality aspect of such designs.     

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.     

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.     

Efficiency balanced designs through reinforcement

In this investigation, general efficiency balanced (GEB) and efficiency balanced (EB) designs with (v + t) treatments, using (i) balanced incomplete block (BIB), (ii) symmetrical BIB, (iii) f -resolvable BIB, (iv) group divisible (GD) and (v) resolvable GD designs have been constructed with smaller number of replications and block sizes.     

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.     

Weakly resolvable IV.3 search designs for the pn factorial experiment

A series of weakly resolvable search designs for the pⁿ factorial experiment is given for which the mean and all main effects are estimable in the presence of any number of two-factor interactions and for which any combination of three or fewer pairs of factors that interact may be detected. The designs have N = p(p–1)n+p runs except in one case where additional runs are required for detection and one case where $\text{(p?1)}\text{2}$ additional runs are needed to estimate all (p–1)² degrees of freedom for each pair of detected interactions. The detection procedure is simple enough that computations can be carried out with hand calculations.     

On resolvable PBIB designs

Resolvable solutions for some two associate PBIB designs obtained by duplicating some non-resolvable designs are given. For the same designs 2-, 3- and 5-resolvable solutions are reported by Clatworthy (1973). A method of construction and some new resolvable PBIB designs obtained through this are given.     

E-optimal incomplete block designs with two distinct block sizes

Sufficient conditions are derived for the determination of E-optimal designs in the class D(v,b₁,b₂,k₁,k₂) of incomplete block designs for v treatments in b₁ blocks of size k₁ each and b₂ blocks of size k₂ each. Some constructions for E-optimal designs that satisfy the sufficient conditions obtained here are given. In particular, it is shown that E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by augmenting b₂ blocks, with k₂ − k₁ extra plots each, of a BIBD(v,b = b₁ + b₂,k₁,λ) and GDD(v,b = b₁ + b₂,k₁,λ₁,λ₂). It is also shown that equireplicate E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by combining disjoint blocks of BIBD(v,b,k₁,λ) and GDD(v,b,k₁,λ₁,λ₂) into larger blocks. As applications of the construction techniques, several infinite series of E-optimal designs with small block sizes differing by at most two are given. Lower bounds for the A-efficiency are derived and it is found that A-efficiency exceeds 99% for v ⩾ 10, and at least 97.5% for 5 ⩽v < 10.     

A survey of nested designs

A nest with parameters (r,k,λ)→(r′,k′,λ′) is a BIBD on (b,v,r,k,λ) where each block has a distinguished sublock of cardinality k, the sublocks forming a (b,v,r,k,λ)-design.These designs are ‘nested’ in the sense of W.T. Federer (1972), who recommended the use of these designs for the sequential addition of periods in marketing experiments in order to retain Youden design properties as rows are added. Note that for a Youden design, the b columns and v treatments are in an SBIBD arrangement with parameters v=b, k=r, and λ.     

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.     

On a class of partially balanced incomplete block designs

Bose and Clatworthy (1955) showed that the parameters of a two-class balanced incomplete block design with λ1=1,λ2=0 and satisfying r <k can be expressed in terms of just three parameters r,k,t. Later Bose (1963) showed that such a design is a partial geometry (r,k,t). Bose, Shrikhande and Singhi (1976) have defined partial geometric designs (r,k,t,c), which reduce to partial geometries when c=0. In this note we prove that any two class partially balanced (PBIB) design with r <k, is a partial geometric design for suitably chosen r,k,t,c and express the parameters of the PBIB design in terms of r,k,t,c and λ2. We also show that such PBIB designs belong to the class of special partially balanced designs (SPBIB) studied by Bridges and Shrikhande (1974).     

An algorithm for constructing optimal resolvable incomplete block designs

This paper describes an efficient algorithm for the construction of optimal or near-optimal resolvable incomplete block designs (IBDs) for any number of treatments v < 100. The performance of this algorithm is evaluated against known lattice designs and the 414 or-designs of Patterson & Williams [36]. For the designs under study, it appears that our algorithm is about equally effective as the simulated annealing algorithm of Venables & Eccleston [42]. An example of the use of our algorithm to construct the row (or column) components of resolvable row-column designs is given.     

Complete designs with blocks of maximal multiplicity

The set of distinct blocks of a block design is known as its support. We construct complete designs with parameters v(?7), k=3, λ=v ? 2 which contain a block of maximal multiplicity and with support size $\text{b}{}^1\text{= (}{}^{\mathrm{v}}{}_3\text{) ? 4(v ? 2)}$. Any complete design which contains such a block, and has parameters v, k, λ as above, must be supported on at most (^v₃) ? 4(v ? 2) blocks. Attention is given to complete designs because of their direct relationship to simple random sampling.     

Some resolvable orthogonal arrays with two symbols

A method of constructing a resolvable orthogonal array (4λk2,2) which can be partitioned into λ orthogonal arrays (4,k 2,1) is proposed. The number of constraints kfor this type of orthogonal array is at most 3λ. When λ=2 or a multiple of 4, an orthogonal array with the maximum number of constraints of 3λ can be constructed. When λ=4n+2(n≧1) an orthogonal array with 2λ+2 constraints can be constructed. When λ is an odd number, orthogonal arrays can be constructed for λ=3,5,7, and 9 with k=4,8,12, and 13 respectively.     

Efficient two-replicate resolvable row-column designs

We relate the efficiency factors of a two-replicate resolvable row-column design to those of a reduced design. This provides a method to search for efficient designs via the reduced design. By choosing the row-component and column-components as generalised cyclic designs, the method is easily implemented and produces efficient designs.     

Some remarks on subdesigns of symmetric designs

Let D(υ, k, λ) be a symmetric design containing a symmetric design D₁(υ₁, k₁, λ₁) (k₁ < k) and let x = υ₁(k ? k₁)/(υ ? υ₁). We show that k ≥(k₁ ? x)² + λ If equality holds, D₁ is called a tight subdesign of D. In the special case, λ₁ = λ, the inequality reduces to that of R.C. Bose and S.S. Shrikhande and tight subdesigns then correspond to their notion of Baer subdesigns. The possibilities for (7upsi;, k, λ) designs having Baer subdesigns are investigated.     

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.     

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.     

A class of resolvable incomplete block designs

A class of resolvable incomplete block designs with three and four replications is obtained here starting from an additive Abelian group.     

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.