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).     

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).     

Computer-Generated Neighbor Designs

Neighbor designs are useful to remove the neighbor effects. In this article, an algorithm is developed and is coded in Visual C + +to generate the initial block for possible first, second,…, and all order neighbor designs. To get the required design, a block (0, 1, 2,…, k ? 1) is then augmented with (v ? 1) blocks obtained by developing the initial block cyclically mod (v ? 1).     

D-optimal designs for covariate models

The problem of finding D-optimal or D-efficient designs in the presence of covariates is considered under a completely randomized design set-up with v treatments, k covariates and N experimental units. In contrast to Lopes Troya [Lopes Troya, J., 1982, Optimal designs for covariates models. Journal of Statistical Planning and Inference, 6, 373–419.], who considered this problem in the equireplicate case, we do not assume that N/v is an integer, and this allows us to study situations where no equireplicate design exists. Even when N/v is an integer, it is seen quite counter-intuitively that there are situations where a non-equireplicate design outperforms the best equireplicate design under the D-criterion.     

Optimal foldover designs

An obvious strategy for obtaining a Doptimal foldover design for p factors at two levels each in 2N runs is to fold a Doptimal main effects plan. We show that this strategy works except when N = 4t + 2 and s is even In that case there are two different classes of D-optimal main effects plans with N runs that have the same determinant. However folding them gives two different values foi the D-optimality criteiion One set of designs is D-optimal The other is not.     

Cyclic Polygonal Designs with Block Size 3 and Joint Distance α = 2

Polygonal designs are useful in survey sampling in terms of balanced sampling plans excluding contiguous units (BSECs) and balanced sampling plans excluding adjacent units (BSAs). In this article, the method of cyclic shifts has been used for the construction of cyclic polygonal designs (in terms of BSAs) with block size k = 3 and λ = 1, 2, 3, 4, 6, 12 for joint distance α = 2 and 51 new designs for treatments v ≤ 100 are given.     

Optimum covariate designs in split-plot and strip-plot design set-ups

The problem considered is that of finding optimum covariate designs for estimation of covariate parameters in standard split-plot and strip-plot design set-ups with the levels of the whole-plot factor in r randomised blocks. Also an extended version of a mixed orthogonal array has been introduced, which is used to construct such optimum covariate designs. Hadamard matrices, as usual, play the key role for such construction.     

Optimal blocked main effects plans

In this paper, we consider the construction of optimal blocked main effects designs where m two-level factors are to be studied in N runs which are partitioned into b blocks of equal size. For N ≡ 2±od4 sufficient conditions are derived for a design to be Φ _f optimal among all designs having main effects occurring equally often at their high and low levels within blocks and then this result is extended to the class of all designs for the case when the block size is two. Methods of constructing designs satisfying the sufficient conditions derived are also 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.     

On the E-optimality of blocked main effects plans when n ≡ 1(mod4)

ABSTRACT

In this article, we consider experimental situations in which m two-lever factors are to be studied using a two-level main effects plan involving n runs which are partitioned into b blocks of size k =?n/b. For the casen ≡ 1(mod?4) and b???k, we derive some new methods of constructing E-optimal designs which tend to be highly efficient under other optimality criteria as well.     

Optimum designs for estimation of regression parameters in a balanced treatment incomplete block design set-up

The use of covariates in block designs is necessary when the experimental errors cannot be controlled using only the qualitative factors. The choice of values of the covariates for a given set-up attaining minimum variance for estimation of the regression parameters has attracted attention in recent times. In this paper, optimum covariate designs (OCD) have been considered for the set-up of the balanced treatment incomplete block (BTIB) designs, which form an important class of test-control designs. It is seen that the OCDs depend much on the methods of construction of the basic BTIB designs. The series of BTIB designs considered in this paper are mainly those as described by Bechhofer and Tamhane (1981) and Das et al. (2005). Different combinatorial arrangements and tools such as Hadamard matrices and different kinds of products of matrices viz Khatri-Rao product and Kronecker product have been conveniently used to construct OCDs with as many covariates as possible.     

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.     

Highly Efficient Designs for Logistic Models with Categorical Variables

Optimal designs for logistic models generally require prior information about the values of the regression parameters. However, experimenters usually do not have full knowledge of these parameters. We propose a design that is D-optimal on a restricted design region. This design assigns an equal weight to design points that contain more information and ignores those design points that contain less information about the regression parameters. The design can be constructed in practice by means of the rank order of the outcome variances. A numerical study compares the proposed design with the D-optimal and completely balanced designs in terms of efficiency.     

A-Optimal and MV-Optimal Incomplete Block Designs for Comparing Successive Treatments

Consider an incomplete block experiment in which observations are taken from t treatments using an incomplete block design with b blocks of size k < t. Suppose the interest is in estimating the differences of effects of successive treatments. This may occur, for example, if the treatments are different dosages or concentrations of a compound. This article presents A-optimal and MV-optimal incomplete block designs for estimating the the differences of successive treatment effects. Tables of optimal designs are given for k < t ≤ 5 with b ≤ 40.     

Optimal Designs for 2 k Choice Experiments

Abstract

In this article we establish the choice sets in the D-optimal design for a choice experiment for testing main effects and for testing main effects and two-factor interactions, when there are k attributes, each with two levels, for choice set size m. We also give a method to construct optimal and near-optimal designs with small numbers of choice sets.     

Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Generalized neighbor designs with block size 3

A generalized neighbor design relaxes the equality condition on the number of times two treatments as neighbors in the design. In this article, we have considered the construction of some classes of generalized neighbor designs with block size k=3 by using the method of cyclic shifts. The distinguishing feature of this construction method is that the properties of a design can easily be obtained from the sets of shifts instead of constructing the actual blocks of the design. A catalog of generalized neighbor designs with block size k=3 is compiled for v∈{5,6,…,18} treatments and for different replications. We provide the reader with a simpler method of construction, and in general the catalog that gives an open choice to the experimenter for selecting any class of neighbor designs.     

Optimum covariate designs in partially balanced incomplete block (PBIB) design set-ups

The use of covariates in block designs is necessary when the covariates cannot be controlled like the blocking factor in the experiment. In this paper, we consider the situation where there is some flexibility for selection in the values of the covariates. The choice of values of the covariates for a given block design attaining minimum variance for estimation of each of the parameters has attracted attention in recent times. Optimum covariate designs in simple set-ups such as completely randomised design (CRD), randomised block design (RBD) and some series of balanced incomplete block design (BIBD) have already been considered. In this paper, optimum covariate designs have been considered for the more complex set-ups of different partially balanced incomplete block (PBIB) designs, which are popular among practitioners. The optimum covariate designs depend much on the methods of construction of the basic PBIB designs. Different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of products of matrices viz. Khatri–Rao product, Kronecker product have been conveniently used to construct optimum covariate designs with as many covariates as possible.     

Locally and maximin optimal designs for multi-factor nonlinear models

This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.