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 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.     

Construction of group divisible designs and rectangular designs from resolvable and almost resolvable balanced incomplete block designs

In this paper, we present a general construction of group divisible designs and rectangular designs by utilising resolvable and “almost resolvable” balanced incomplete block designs. As special cases, we obtain the following E-optimal designs: (a) Group divisible (GD) designs with λ₂=λ₁+1 and (b) Rectangular designs with 2 rows and having λ₃=λ₂−1=λ₁+1. Many of the GD designs are optimal among binary designs with regard to all type 1 criteria.     

Resolvable systems of 8-tuples

Let R be a family of k-element blocks of a v-element set V such that any two elements of V are contained in λ blocks of R and R=R₁∪…∪R_v?1, R_i∩ R_j=? (i≠j) and ?{B_i∈R_j∣i=1,…,v?k}=V (B_i a block in R_j), i.e. R is a resolvable block design RB(v, k, λ). In this paper it will be shown that a sufficient condition for the existence of an RB(v, 8, 7) is that v≡0 (mod 8) and v is nondivisible by 3, 5, 7.     

MBMUDs: a combinatorial extension of BIBDs showing good optimality behaviour

The construction of a balanced incomplete block design (BIBD) is formulated in terms of combinatorial optimization by defining a cost function that reaches its lower bound on all and only those configurations corresponding to a BIBD. This cost function is a linear combination of distribution measures for each of the properties of a block design (number of plots, uniformity of rows, uniformity of columns, and balance). The approach generalizes naturally to a super-class BIBDs, which we call maximally balanced maximally uniform designs (MBMUDs), that allow two consecutive values for their design parameters [r,r+1;k,k+1;λ,λ+1]. In terms of combinatorial balance, MBMUDs are the closest possible approximation to BIBDs for all experimental settings where no set of admissible parameters exists. Thus, other design classes previously proposed with the same approximation aim—such as RDGs, SRDGs and NBIBDs of type I—can be viewed as particular cases of MBMUDs. Interestingly, experimental results show that the proposed combinatorial cost function has a monotonic relation with A- and D-statistical optimality in the space of designs with uniform rows and columns, while its computational cost is much lower.     

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.     

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 block designs arising from rank 3 graphs

In this paper a criterion showing when the orbit of a subgraph of a given rank 3 graph forms a block design, is given. As an application several classes of block designs are derived from the triangular graph T(n) and the lattice graph L₂(n).     

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.     

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.     

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.     

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).     

On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1

The norm $\text{6A6 = {tr(A′A)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for 6A6 will be given for a balanced fractional 2^m factorial design of resolution 2l + 1 which obtained from a simple array with parameters (m; λ₀, λ₁,…, λ_m). This array is identical with a balanced array of strength m, m constraints and index set {λ₀, λ₁,…, λ_m}. In the class of the designs of resolution V (l = 2) obtained from S-arrays, ones which minimize 6A6 will be presented for any fixed N assemblies satisfying (i) m = 4, 11 ? N ? 16, (ii) m = 5, 16 ? N ? 32, and (iii) m = 6, 22 ? N ? 40.     

ON INCOMPLETE BLOCK DESIGNS FOR SYMMETRIC PARALLEL LINE ASSAYS

Incomplete block designs for symmetric parallel line assays, which estimate L_p, L₁and L1/1 with full efficiency, are constructed in a unified manner. Necessary and sufficient conditions are given for existence. A complete solution is presented when the number of doses is even. A general systematic method of construction is given when the number of doses is odd, and all possible designs for up to 15 doses of each preparation are tabulated.     

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.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block 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).     

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).     

Constructions of Nested Pairwise Efficiency and Variance Balanced Designs

ABSTRACT

Nested pairwise efficiency and variance balanced designs form a new class of block designs. In this article, two methods of constructing such designs from a symmetric balanced incomplete block design are proposed with some illustrations.     

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.