Randomization of neighbour balanced generalized Youden designs

If a crossover design with more than two treatments is carryover balanced, then the usual randomization of experimental units and periods would destroy the neighbour structure of the design. As an alternative, Bailey [1985. Restricted randomization for neighbour-balanced designs. Statist. Decisions Suppl. 2, 237–248] considered randomization of experimental units and of treatment labels, which leaves the neighbour structure intact. She has shown that, if there are no carryover effects, this randomization validates the row–column model, provided the starting design is a generalized Latin square. We extend this result to generalized Youden designs where either the number of experimental units is a multiple of the number of treatments or the number of periods is equal to the number of treatments. For the situation when there are carryover effects we show for so-called totally balanced designs that the variance of the estimates of treatment differences does not change in the presence of carryover effects, while the estimated variance of this estimate becomes conservative.     

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.     

ON THE DESIGN OF FIELD EXPERIMENTS UNDER A ONE-DIMENSIONAL NEIGHBOUR MODEL

This paper examines the effect of randomisation restrictions, either to satisfy conditions for a balanced incomplete block design or to attain a higher level of partial neighbour balance, on the average variance of pair-wise treatment contrasts under a neighbour model discussed by Gleeson & Cullis (1987). Results suggest that smaller average pairwise variances can be obtained by ignoring requirements for incomplete block designs and concentrating on achieving a higher level of partial neighbour balance. Field layout of the design, although often determined by practical constraints, e.g. size, shape of site, minimum plot size and experimental husbandry, may markedly affect average pairwise variance. For the one-dimensional (row-wise) neighbour model considered here, investigation of three different layouts suggests that for a rectangular array of plots, smaller average pairwise variances can generally be obtained from layouts with fewer rows and more plots per row.     

Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

Efficient circular neighbour designs for spatial interference model

We consider circular block designs for field-trials when there are two-sided spatial interference between neighbouring plots of the same blocks. The parameter of interest is total effects that is the sum of direct effect of treatment and neighbour effects, which correspond to the use of a single treatment in the whole field. We determine universally optimal approximate designs. When the number of blocks may be large, we propose efficient exact designs generated by a single sequence of treatment. We also give efficiency factors of the usual binary block neighbour balanced designs which can be used when the number of blocks is small.     

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.     

Experimental designs for open pollination in polycross trials

A polycross is the pollination by natural hybridization of a group of genotypes, generally selected, grown in isolation from other compatible genotypes in such a way to promote random open pollination. A particular practical application of the polycross method occurs in the production of a synthetic variety resulting from cross-pollinated plants. Laying out these experiments in appropriate designs, known as polycross designs, would not only save experimental resources but also gather more information from the experiment. Different situations may arise in polycross nurseries where accordingly different polycross designs may be used. For situations in which some genotypes interfere in the growth or production of other genotypes, but have to be grown together, neighbour-restricted design is a better option. Furthermore, when the topography of the nursery is such that a known wind system in a certain direction may prevail, then designs balanced for neighbour effects of genotypes only in the direction of wind are appropriate which may help in saving experimental resources to a great extent. Also, when genotypes are planted in a small area without leaving much space between rows, designs balanced for neighbour effects from all possible eight directions are useful to have equal chance of pollinating and being pollinated by every other genotype. Here, polycross designs have been obtained to match above-mentioned three situations. SAS Macros have also been developed to generate these proposed designs.     

Regular E-Optimal Spring Balance Weighing Designs with Correlated Errors

The problems linked with an E-optimal spring balance weighing design with correlated errors are discussed. The topic is focus on the determining the maximal eigenvalue of the inverse of the information matrix of estimators. The constructing method of the E-optimal design, based on the incidence matrices of balanced incomplete block designs, is presented.     

Row-column designs for comparing treatments with a control

This paper considers the problem of the design and analysis of experiments for comparing several treatments with a control when heterogeneity is to be eliminated in two directions. A class of row-column designs which are balanced for treatment vs. control comparisons (referred to as the balanced treatment vs. control row-column or BTCRC designs) is proposed. These designs are analogs of the so-called BTIB designs proposed by Bechhofer and Tamhane (Technometrics 23 (1981) 45–57) for eliminating heterogeneity in one direction. Some methods of analysis and construction of these designs are given. A measure of efficiency of BTCRC designs in terms of the A-optimality criterion is derived and illustrated by several examples.     

Low-order spatially-distinct latin squares

The order three to five spatially-distinct Latin squares, and the order three to six spatially-distinct Latin square treatment designs are listed. Some statistical results are given. Designs for 4, 5 and 6 treatments that were found previously to be robust to a linear by linear interacrion are shown to be optimal. Designs with good neighbour balanced are also considered.     

A Note on Change-Over Designs

A change-over model with correlated errors is discussed. In particular the results of Patterson (1950) and Lucas (1951) for balanced change-over designs are extended to more than four treatments and more general variance matrices. A connection with recent work on neighbour models for field trials is made.     

Use of balanced bipartite weighing designs as chemical balance designs

This paper deals with the use of balanced bipartite weighing designs as chemical balance designs when there Is a restriction on the number of objects that can be placed on either pan..The variance of each estimated weight and the covarlance of each pair of estimated weights are worked out.The use of additional weighing when the design matrix is singular is discussed.     

EFFICIENT BLOCK DESIGNS FOR CORRELATED OBSERVATIONS

Current methods for the design of efficient incomplete block experiments when the observations within a block are dependent usually involve computer searches of binary designs. These searches give little insight into the features that lead to efficiency, and can miss more efficient designs. This paper aims to develop some approximations which give some insight into the features of a design that lead to high efficiency under a generalized least-squares analysis for a known dependence structure, and to show that non-binary designs can be more efficient for some dependence structures. In particular, we show how neighbour balance and end plot balance are related to the design efficiency for low-order autoregressions, and that under moderate positive dependence, replication at lag two can sometimes increase efficiency.     

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.     

PARTIALLY EQUINEIGHBOURED DESIGNS

This paper presents further results on a class of designs called equineighboured designs, ED. These designs are intended for field and related experiments, especially whenever there is evidence that observations in the same block are correlated. An ED has the property that every unordered pair of treatments occurs as nearest neighbours equally frequently at each level. Ipinyomi (1986) has defined and shown that ED are balanced designs when neighbouring observations are correlated. He has also presented ED as a continuation of the development of optimal block designs. An ED would often require many times the number of experimental materials needed for the construction of an ordinary balanced incomplete block, BIB, design for the same number of treatments and block sizes. Thus for a relatively large number of treatments and block sizes the required minimum number of blocks may be excessively large for practical use of ED. In this paper we shall define and examine partially equineighboured designs with n concurrences, PED (n), as alternatives where ED are practically unachievable. Particular attention will be given to designs with smaller numbers of blocks and for which only as little balance as possible may be lost.     

D-optimal weighing designs and optimal chemical balance weighing designs for estimating the total weight

This paper provides D-optimal spring balance designs for estimating individual weights when the number of objects to be weighed in each weighing, B, is fixed. D-optimal chemical balance designs for estimating total weight under both homogeneous and nonhomogeneous error variances are found when the number of objects weighed in each weighing is ≥ B, a fixed number.

We indicate the restriction used in Chacko & Dey(1978) and Kageyama(1988), i.e. that chemical designs X be restricted to designs in which exactly “a” objects are replaced on the left pan and exactly “b” on the right pan in each of the weighings for a, b > 0, is unnecessary.     

Optimal Designs for the Carryover Model with Random Interactions Between Subjects and Treatments

The paper considers a model for crossover designs with carryover effects and a random interaction between treatments and subjects. Under this model, two observations of the same treatment on the same subject are positively correlated and therefore provide less information than two observations of the same treatment on different subjects. The introduction of the interaction makes the determination of optimal designs much harder than is the case for the traditional model. Generalising the results of Bludowsky's thesis, the present paper uses Kushner's method to determine optimal approximate designs. We restrict attention to the case where the number of periods is less than or equal to the number of treatments. We determine the optimal designs in the important special cases that the number of periods is 3, 4 or 5. It turns out that the optimal designs depend on the variance of the random interactions and in most cases are not binary. However, we can show that neighbour balanced binary designs are highly efficient, regardless of the number of periods and of the size of the variance of the interaction effects.     

On the analysis of unbalanced two-level supersaturated designs via generalized linear models

Supersaturated designs (SSDs) are factorial designs in which the number of experimental runs is smaller than the number of parameters to be estimated in the model. While most of the literature on SSDs has focused on balanced designs, the construction and analysis of unbalanced designs has not been developed to a great extent. Recent studies discuss the possible advantages of relaxing the balance requirement in construction or data analysis of SSDs, and that unbalanced designs compare favorably to balanced designs for several optimality criteria and for the way in which the data are analyzed. Moreover, the effect analysis framework of unbalanced SSDs until now is restricted to the central assumption that experimental data come from a linear model. In this article, we consider unbalanced SSDs for data analysis under the assumption of generalized linear models (GLMs), revealing that unbalanced SSDs perform well despite the unbalance property. The examination of Type I and Type II error rates through an extensive simulation study indicates that the proposed method works satisfactorily.     

Construction of circular partially balanced-Repeated measurement designs using cyclic shifts

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. These designs balance out the residual effects. The situations where balanced repeated measurements designs require a large number of the subjects, partially-balanced repeated measurements designs should be used. In this paper some infinite series are developed which provide circular partially-balanced repeated measurement designs for p (periods) even. Catalogues of circular partially-balanced repeated measurement designs are also presented for v (treatments) ≤ 100 with p = 5, 7 & 9.     

Crossover designs in the presence of carry-over effects from two factors

Experiments, used in the telecommunications industry and elsewhere, are considered that involve the simultaneous application of levels of two unrelated factors, treatments and stimuli, to each of several subjects in a succession of time periods. The existence is suspected of carry-over effects of treatments and stimuli, in the period immediately following the period of their application. Methods are given for the construction of separate sequences of treatments and of stimuli; these methods are based on the Latin squares of Williams and of Russell. In the resulting designs, the treatments and stimuli are either orthogonal or nearly orthogonal, and the coincidence of the direct and carry-over effects of each factor is either balanced or nearly balanced. The efficiencies of the designs are assessed by comparing the average variances of elementary contrasts in the levels of each factor with appropriate lower bounds.