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).     

E-optimal design in irregular BIBD settings

When the necessary conditions for a BIBD are satisfied, but no BIBD exists, there is no simple answer for the optimal design problem. This paper identifies the E-optimal information matrices for any such irregular BIBD setting when the number of treatments is no larger than 100. A- and D-optimal designs are typically not E-optimal. An E-optimal design for 15 treatments in 21 blocks of size 5 is found.     

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.     

Many (22, 44, 14, 7, 4) — and (15, 42, 14, 5, 4)-balanced incomplete block designs

A Balanced Incomplete Block Design (BIBD) is a pair (V, B) where V is a v-set and B is a collection of b k-subsets of V, called blocks, such that every element of V occurs in exactly r of the k-subsets and every 2-subset of V occurs in exactly λ of the blocks. The number of non-isomorphic designs of a BIBD (22, 44, 14, 7, 4) whose automorphism group is divisible by 7 or 11 are investigated. From this work, results are obtained on the number of non-isomorphic BIBDs (15, 42, 14, 5, 4).     

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.     

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.     

Optimal regular graph designs

A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. The most popular criteria include the A- and D-criteria. Regular graph designs occur in many optimality results, and if the number of blocks is large enough, an A-optimal (or D-optimal) design is among them (if any exist). To explore the landscape of designs with a large number of blocks, we introduce extensions of regular graph designs. These are constructed by adding the blocks of a balanced incomplete block design repeatedly to the original design. We present the results of an exact computer search for the best regular graph designs and the best extended regular graph designs with up to 20 treatments v, block size \(k \le 10\) and replication r \(\le 10\) and \(r(k-1)-(v-1)\lfloor r(k-1)/(v-1)\rfloor \le 9\).     

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.     

Optimal designs,supplementary difference sets and multipliers

We investigate multipliers of 2 - {v; q², q²; λ} supplementary difference sets where cyclotomy has been used to construct D-optimal designs.     

D-Optimal Designs for Covariate Parameters in Block Design Set Up

The problem considered is that of finding D-optimal design for the estimation of covariate parameters and the treatment and block contrasts in a block design set up in the presence of non stochastic controllable covariates, when N = 2(mod 4), N being the total number of observations. It is clear that when N ≠ 0 (mod 4), it is not possible to find designs attaining minimum variance for the estimated covariate parameters. Conditions for D-optimum designs for the estimation of covariate parameters were established when each of the covariates belongs to the interval [?1, 1]. Some constructions of D-optimal design have been provided for symmetric balanced incomplete block design (SBIBD) with parameters b = v, r = k = v ? 1, λ =v ? 2 when k = 2 (mod 4) and b is an odd integer.     

COMPUTER-AIDED CONSTRUCTION OF SMALL (M, S)-OPTIMAL INCOMPLETE BLOCK DESIGNS

A new exchange algorithm for the construction of (M, S)-optimal incomplete block designs (IBDS) is developed. This exchange algorithm is used to construct 973 (M, S)-optimal IBDs (v, k, b) for v= 4,…,12 (varieties) with arbitrary v, k (block size) and b (number of blocks). The efficiencies of the “best” (M, S)-optimal IBDs constructed by this algorithm are compared with the efficiencies of the corresponding nearly balanced incomplete block designs (NBIBDs) of Cheng(1979), Cheng & Wu (1981) and Mitchell & John(1976).     

Algorithmic and analytical construction of efficient designs in small blocks for comparing consecutive pairs of treatments

Optimal block designs in small blocks are explored under the A-, E- and D-criteria when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first formulate the problem via approximate theory which leads to a convenient multiplicative algorithm for obtaining A-optimal design measures. This, in turn, yields highly efficient exact designs, under the A-criterion, even when the number of blocks is rather small. Moreover, our approach is seen to allow nesting of such efficient exact designs which is an advantage when the resources for the experiment are available in possibly several stages. Illustrative examples are given and tables of A-optimal design measures are provided. Approximate theory is also seen to yield analytical results on E- and D-optimal design measures.     

Optimal block designs comparing treatments with a control when the errors are correlated

The efficient design of experiments for comparing a control with v new treatments when the data are dependent is investigated. We concentrate on generalized least-squares estimation for a known covariance structure. We consider block sizes k equal to 3 or 4 and approximate designs. This method may lead to exact optimal designs for some v, b, k, but usually will only indicate the structure of an efficient design for any particular v, b, k, and yield an efficiency bound, usually unattainable. The bound and the structure can then be used to investigate efficient finite designs.     

E- and R-optimality of block designs for treatment-control comparisons

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple linear constraints. Based on this characterization, we obtain a class of E-optimal exact designs for unequal block sizes. In the studied model, we provide a statistical interpretation for wide classes of E-optimal designs. Moreover, we show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.     

Optimal block designs for three treatments when observations are correlated

E$E$-optimal designs for comparing three treatments in blocks of size three are identified, where intrablock observations are correlated according to a first order autoregressive error process with parameter ρ∈(0,1)$\rho \in (0,1)$. For number of blocks b   of the form b=3n+1$b=3n+1$, there are two distinct optimal designs depending on the value of ρ$\rho$, with the best design being unequally replicated for large ρ$\rho$. For other values of b$b$, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established.     

Addition of runs to a two-level supersaturated design

The purpose of this article is to introduce a new class of extended E(s²)-optimal two level supersaturated designs obtained by adding runs to an existing E(s²)-optimal two level supersaturated design. The extended design is a union of two optimal SSDs belonging to different classes. New lower bound to E(s²) has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(s²)-optimal SSDs are also included.     

Concerning the vector v(m,t)

Let q = mt + 1 be a prime power, and let v(m, t) be the (m + 1)-vector (b₁, b₂, …, b_{m + 1}) of elements of GF(q) such that for each k, 1 ⩽ k ⩽ m + 1, the set {b_i−b_j:i∈{1,2,…m+1} − {m + 2 − k}, j  i + k(mod m + 2) and 1⩽j⩽m+1} forms a system of representatives for the cyclotomic classes of index m in GF(q). In this paper, we investigate the existence of such vectors. An upper bound on t for the existence of a v(m, t) is given for each fixed m unless both m and t are even, in which case there is no such a vector. Some special cases are also considered.     

Approximate E-optimal designs for the model of spring balance weighing with a constant bias

The present paper analyzes the linear regression model with a nonzero intercept term on the vertices of a d-dimensional unit cube. This setting may be interpreted as a model of weighing d objects on a spring balance with a constant bias. We give analytic formulas for E-optimal designs, as well as their minimal efficiencies under the class of all orthogonally invariant optimality criteria, proving the criterion-robustness of the E-optimal designs. We also discuss the D- and A-optimal designs for this model.