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 a class of variance balanced incomplete block designs

In this paper variance balanced incomplete block designs have been constructed for situations when suitable BIB designs do not exist for a given number of treatments, because of the contraints bk=vr, λ(v-1) = r(k-l). These variance balanced designs are in unequal block sizes and unequal replications.     

A note on generalized binary proper efficiency-balanced block designs

Some methods of construction of generalized binary proper efficiency-balanced block (GB-EB) designs with two different replications are derived and the solution of the unknown designs given by Das and Kageyama (1991) are obtained.     

Construction of Balanced Incomplete Block Designs Using Cyclic Shifts

Balanced incomplete block designs (BIBDs) play important role in design of experiments, especially in field experiments. These designs ensure that treatments are compared with equal precision. Several methods are available in the literature to construct BIBDs but in this article some infinite series of these designs are presented by method of cyclic shifts. This method expresses some standard properties of a design just through examining the sets of shifts rather than studying the whole design.     

A basic lemma and the analysis of block and Kronecker product designs

In this paper the analysis of the class of block designs whose C matrix can be expressed in terms of the Kronecker product of some elementary matrices is considered. The analysis utilizes a basic result concerning the spectral decomposition of the Kronecker product of symmetric matrices in terms of the spectral decomposition of the component matrices involved in the Kronecker product. The property (A) of Kurkjian and Zelen (1963) is generalised and the analysis of generalised property (A) designs is given. It is proved that a design is balanced factorially if and only if it is a generalised property (A) design. A method of analysis of Kronecker product block designs whose component designs are equi-replicate and proper is also suggested.     

Comparison of designs equivalent under one or two criteria

Two designs equivalent under one or two criteria may be compared under other criteria. For certain configurations of eigenvalues of the information matrices, we decide which design is the better of the two for many other such criteria. The relationship to universal optimality (in the case of equivalence under one criterion) is indicated. For two criteria, applications are given to weighing and treatment-with-covariate settings.     

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.     

BALANCED ARRAYS AND WEIGHING DESIGNS

Dey (19711, Saha (1975), Kageyama & Saha (1983) and others have shown how optimum chemical balance weighing designs can be constructed from the incidence matrices of balanced incomplete block (BIB) designs. In this paper, it is shown that weighing designs can be constructed from some suitably chosen two-symbol balanced arrays of strength two, which need not always be incidence matrices of BIB designs. The findings lead us to construct new optimum chemical balance weighing designs from incidence matrices of BIB designs.     

Lower Bounds for the Efficiency of Designs with Respect to the D-Criterion when the Observations are Correlated

In the general linear model consider the experimental design problem for the Gauß-Markov estimator or least squares estimator when the observations are correlated. We prove new formulas for the efficiency of an exact design with respect to the D-criterion. For models with intercept term, for example, these formulas are useful to derive better lower bounds for the efficiency than the bounds recently given for an arbitrary linear model. These bounds are applied in examples to symmetrical regular circulants as covariance matrices. A byproduct of the investigations is some insight as to what kinds of designs might retain their optimality or high efficiency (for the uncorrelated homoscedastic case) under correlated observations.     

The analysis of mixed and random effects models for equireplicate variance-balanced block designs

Equirephcate variance-balanced block designs possess many desirable properties and include several important subclasses of designs, such as balanced incomplete block designs. However, while attention has been paid to the derivation of properties and the construction of these designs, methods of analyzing experiments using these designs have not been developed completely. Thus, we consider models appropriate to these designs, construct test statistics for basic hypotheses and derive the distributions for these statistics. Four additive models that include all of the possible combinations of fixed and random effects are examined. The analyses are generalizations of the analyses for balanced incomplete block designs, and include the intra-block, intra-treatment and inter-block analyses, Estimators of fixed treatment effects and their distributions also are given.     

Bayesian D-optimal designs for exponential regression models

We consider the Bayesian D-optimal design problem for exponential growth models with one, two or three parameters. For the one-parameter model conditions on the shape of the density of the prior distribution and on the range of its support are given guaranteeing that a one-point design is also Bayesian D-optimal within the class of all designs. In the case of two parameters the best two-point designs are determined and for special prior distributions it is proved that these designs are Bayesian D-optimal. Finally, the exponential growth model with three parameters is investigated. The best three-point designs are characterized by a nonlinear equation. The global optimality of these designs cannot be proved analytically and it is demonstrated that these designs are also Bayesian D-optimal within the class of all designs if gamma-distributions are used as prior distributions.     

A-Optimal Block Designs with Additional Singly Replicated Treatments

Block designs to which have been added a number of singly-replicated treatments, known as secondary treatments, are particularly useful for experiments where only small amounts of material are available for some treatments, for example new plant varieties. The designs are of particular use in the microarray situation. Such designs are known as 'augmented designs'. This paper obtains the properties of these designs and shows that, with an equal number of secondary treatments in each block, the A-optimal design is obtained by using the A-optimal design for the original block design. It develops formulae for the variance of treatment comparisons, for both the primary and the secondary treatments. A number of examples are used to illustrate the results.     

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.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

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).     

Optimal designs for generalized linear models

Summary This paper solves some D-optimal design problems for certain Generalized Linear Models where the mean depends on two parameters and two explanatory variables. In all of the cases considered the support point of the optimal designs are found to be independent of the unknown parameters. While in some cases the optimal design measures are given by two points with equal weights, in others the support is given by three point with weights depending on the unknown parameters, hence the designs are locally optimal in general. Empirical results on the efficiency of the locally optimal designs are also given. Some of the designs found can also be used for planning D-optimal experiments for the normal linear model, where the mean must be positive. This research was carried out in part at University College, London as an M.Sc. project. Thanks are due to Prof. I. Ford (University of Glasgow) and Prof. A. Giovagnoli (University of Perugia) for their valuable suggestions and critical observations.     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

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.     

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.     

Optimal discrimination designs for exponential regression models

We investigate optimal designs for discriminating between exponential regression models of different complexity, which are widely used in the biological sciences; see, e.g., Landaw [1995. Robust sampling designs for compartmental models under large prior eigenvalue uncertainties. Math. Comput. Biomed. Appl. 181–187] or Gibaldi and Perrier [1982. Pharmacokinetics. Marcel Dekker, New York]. We discuss different approaches for the construction of appropriate optimality criteria, and find sharper upper bounds on the number of support points of locally optimal discrimination designs than those given by Caratheodory's Theorem. These results greatly facilitate the numerical construction of optimal designs. Various examples of optimal designs are then presented and compared to different other designs. Moreover, to protect the experiment against misspecifications of the nonlinear model parameters, we adapt the design criteria such that the resulting designs are robust with respect to such misspecifications and, again, provide several examples, which demonstrate the advantages of our approach.