Approximate theory of multiway block designs

Optimality properties of multiway block designs are deduced from the general results of J. Kiefer's approximate-design theory. In the model with additive effects these optimality properties solely depend on the two-dimensional marginals of the designs. Uniform designs, and designs whose two-dimensional marginals are products of the one-dimensional marginals, are shown to be optimal. Approximate Youden designs are introduced for the case when the support sets of the two-dimensional marginals are prescribed in advance. They are optimal in a relatively small class of competing designs only.     

Optimal block designs for triallel cross experiments

Optimal block designs for a certain type of triallel cross experiments are investigated. Nested balanced block designs are introduced and it is shown how these designs give rise to optimal designs for triallel crosses. Several .series of nested balanced block designs, leading to optimal designs for triallel crosses are reported.     

E-optimal block designs under heteroscedastic model

This paper mainly studies the E-optimality of block designs under a general heteroscedastic setting. The C-matrix of a block design under a heteroscedastic setting is obtained by using generalized least squares. Some bounds for the smallest positive eigenvalue of C-matrix are obtained in some general classes of connected designs. Use of these bounds is then made to obtain certain E-optimal block designs in various classes of connected block designs.     

A note on trend-free block designs

Trend-free and nearly trend-free block designs were developed to eliminate polynomial trends across the plots of experimental designs. Yeh, Bradley and Notz (1985) proved that certain nearly trend-free designs are A- and D-optimal in a subclass of all competing designs. This article extends that result by enlarging the class of designs for which the optimality holds, and by increasing the class of optimality criteria from A- and D-optimality to the class of all Schur-convex nonincreasing functions.     

Resolvable block designs for factorial experiments

It is known that n-cyclic designs provide a flexible class of designs suitable for setting out factorial experiments. In this paper we show that many of these designs are resolvable. Further, an extensive class of practically useful designs can be derived from them by deleting replicates. The properties of the designs compare favourably with those obtained by the algorithm of Williams and John (1996) (Appl. Statist. 45, 39–46).     

Application of minque procedure to block designs

Method of minimum norm quadratic unbiased estimation (MINQUE) is applied to incomplete block designs. Simple formulae are derived for a class of designs which includes the balanced designs.     

On nested block designs geometry

Nested block designs and block designs properties such as orthogonality, orthogonal block structure and general balance are examined using the concept of a commutative quadratic subspace and standard properties of orthogonal projectors. In this geometrical context conditions for existence of the best linear unbiased estimators of treatment contrasts are also discussed.     

Optimal regression designs in the presence of random block effects

A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.     

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.     

On Kronecker block designs for factorial experiments

In this paper the use of Kronecker designs for factorial experiments is considered. The two-factor Kronecker design is considered in some detail and the efficiency factors of the main effects and interaction in such a design are derived. It is shown that the efficiency factor of the interaction is at least as large as the product of the efficiency factors of the two main effects and when both the component designs are totally balanced then its efficiency factor will be higher than the efficiency factor of either of the two main effects. If the component designs are nearly balanced then its efficiency factor will be approximately at least as large as the efficiency factor of either of the two main effects. It is argued that these designs are particularly useful for factorial experiments.Extensions to the multi-factor design are given and it is proved that the two-factor Kronecker design will be connected if the component designs are connected.     

A class of universally optimal binary block designs

Consider block designs with block size of one less than the number of treatments. It is shown that a universally optimal design over the binary class always exists. The construction of these designs is simple. Optimality properties of these designs over the global class are discussed.     

Balanced block designs

In the present paper an attempt has been made to characterize and unify the three different concepts of balancing in incomplete block designs, namely (i) variance balance, (ii) efficiency balance and (iii) pairwise balance. Simple characterizations of variance balance and efficiency balance have been given using the P matrix. A method of constructing efficiency balanced (EB) and variance balanced designs has also been presented.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

Characterization of certain incomplete block designs

It is shown that certain inequalities known for partially balanced incomplete block (PBIB) designs remain valid for general incomplete block designs. Some conditions for attaining their bounds are also given. Furthermore, the various types of PBIB designs are characterized by relating blocks of designs with association schemes. The approach here is based on the spectral expansion of NN' for the incidence matrix N of an incomplete block design.     

Optimal saturated block designs when observations are correlated

A- and MV-optimal block designs are identified in the class of minimally connected designs when the observations within blocks are spatially correlated. All connected designs are shown to be D-equal regardless of the correlation structure, and a sufficient condition for E-optimality is presented. Earlier results for the uncorrelated case are strengthened.     

Inequality for equireplicated n-ary block designs with unequal block sizes

It is shown that certain inequalities known for binary, equireplicated, equiblock-sized block designs remain valid for equireplicated n-ary block designs with unequal block sizes. The approach used here is based on the spectral expansion of the C-matrix of the block design. The main theorems include some useful and combinatorially interesting results.     

On the existence of linear trend free block designs

We present a class of counerexamples for a conjecture on the existence or linear trend free block designs we will also prove a considerably weakened version of this conjecture which will determine all combinations of designs parmetres for which the class of linear trend free block designs is non empty.     

On construction of constant block-sum partially balanced incomplete block designs

Abstract

Constant block-sum designs are of interest in repeated measures experimentation where the treatments levels are quantitative and it is desired that at the end of the experiments, all units have been exposed to the same constant cumulative dose. It has been earlier shown that the constant block-sum balanced incomplete block designs do not exist. As the next choice, we, in this article, explore and construct several constant block-sum partially balanced incomplete block designs. A natural choice is to first explore these designs via magic squares and Parshvanath yantram is found to be especially useful in generating designs for block size 4. Using other techniques such as pair-sums and, circular and radial arrangements, we generate a large number of constant block-sum partially balanced incomplete block designs. Their relationship with mixture designs is explored. Finally, we explore the optimization issues when constant block-sum may not be possible for the class of designs with a given set of parameters.     

Optimal balanced block designs for correlated observations

The construction of universally optimal designs, if such exist, is difficult to obtain, especially when there are some nuisance effects or correlated errors. The hub correlation is a special correlation structure with applications to experiments in genetics, networks and other areas in industry and agriculture. There may be restrictions on the correlation values of the hub structure depending on the experiment. Optimality of block designs under hub correlation has been studied for the case of a constant correlation value. In this article, we consider the hub structure when one of the correlation values is different from the others, and the universally optimal block designs, binary or non-binary, are theoretically obtained. Also, we introduce an algorithm to construct the optimal designs. The Canadian Journal of Statistics 48: 596–604; 2020 © 2020 Statistical Society of Canada     

Block designs with block size two

Algorithms are given for the construction of binary block designs with replications and concurrences differing by at most one. The designs are resolvable and/or connected wherever the parameters permit.