On the Construction of Nearest–Neighbour Balanced Row–Column Designs

Nearest–neighbour balance is considered a desirable property for an experiment to possess in situations where experimental units are influenced by their neighbours. This paper introduces a measure of the degree of nearest–neighbour balance of a design. The measure is used in an algorithm which generates nearest–neighbour balanced designs and is readily modified to obtain designs with various types of nearest–neighbour balance. Nearest–neighbour balanced designs are produced for a wide class of parameter settings, and in particular for those settings for which such designs cannot be found by existing direct combinatorial methods. In addition, designs with unequal row and column sizes, and designs with border plots are constructed using the approach presented here.     

Minimal circular balanced repeated measurement designs in unequal period sizes

Abstract

Repeated measurement designs (RMDs) are widely used in medicine, pharmacology, animal sciences and psychology. In these fields, there are several situations where these designs should be used in periods of different sizes. With the use of RMD, residual effects or carry over effects may arise and balanced RMDs are solution to this problem. In this article, therefore, some infinite series are developed through method of cyclic shifts to obtain circular balanced repeated measurements designs in periods of two different sizes.     

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.     

Some new augmented fractional Box–Behnken designs

The augmented Box–Behnken designs are used in the situations in which Box–Behnken designs (BBDs) could not estimate the response surface model due to the presence of third-order terms in the response surface models. These designs are too large for experimental use. Usually experimenters prefer small response surface designs in order to save time, cost, and resources; therefore, using combinations of fractional BBD points, factorial design points, axial design points, and complementary design points, we augment these designs and develop new third-order response surface designs known as augmented fractional BBDs (AFBBDs). These AFBBDs have less design points and are more efficient than augmented BBDs.     

Some new construction of circular weakly balanced repeated measurements designs in periods of two different sizes

Abstract

Balanced repeated measurements designs (RMDs) balance out the residual effects. Williams Latin square designs work as minimal combinatorial balanced as well as variance balanced for RMDs for p (period sizes) = v (number of treatments). If minimal balanced RMDs cannot be constructed for the situations where p must be less than v then weakly balanced RMDs should be preferred. In this article, some generators are developed to generate circular weakly balanced RMDs in periods of two different sizes. To obtain the proposed designs, some construction procedures are also described for some of the cases where we could not develop generators.     

Non binary partially neighbor balanced designs for circular blocks

ABSTRACT

Neighbor designs are recommended for the cases where the performance of treatment is affected by the neighboring treatments as in biometrics and agriculture. In this paper we have constructed two new series of non binary partially neighbor balanced designs for v = 2n and v = 2n+1 number of treatments, respectively. The blocks in the design are non binary and circular but no treatment is ever a neighbor to itself. The designs proposed here are partially balanced in terms of nearest neighbors. No such series are known in the literature.     

Optimal repeated measurements designs for comparing test treatments with a control

A-optimal and mv optimal repeated measurments designs for comparing serveral test treatments with a control are considered. the models considered are basically of two types: without preperides and the cirular model. It is shown known that some known strongly balanced uniform repeated measurements designs can be modified to obtain optimal designs for this problem. Some other methods of finding optimal designs are also given.     

Partially efficiency-balanced designs-II

efficiency balanced (£R) and partially efficiency balanced (PEB) designs were studied hy Calinski (1971) and Puri and Higam (1971, 1977a). In tais paper we have studied the dual desinas of two classes of PEB designs. It is shown that the dual of any symmetrical PCS design with HN'=N'H is again a PEB design. Also the dual of a PEB(S) with μ₁ = 0 μ₂ = 0and is shown to the again a PEB(S). The relations between PEB designs and linked and partially linked blocks have heen established. As a conseguence, the analysis of linked Mock and partially linkei block designs can he sinrlififid. Sore useful classes of supplencntcd block designs having simple analysis have also been discussed.     

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.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

On balanced incomplete Latin square designs

Recently, balanced incomplete Latin square designs are introduced in the literature. We propose three methods of constructions of balanced incomplete Latin square designs. Particular classes of Latin squares namely Knut Vik designs, semi Knut Vik designs, and crisscross Latin squares play a key role in the construction.     

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.     

Nonproper variance balanced designs and optimality

This communication deals with the construction and optimality of non-proper (unequal block sized) variance balanced (VB) designs obtainable under linear homoscedastic normal model. Several methods of construction of non-proper VB designs have been given. Some constructed designs are universally optimal non-proper variance balanced designs.     

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.     

The performance of subset response surface designs for estimating third order terms

Response surface designs are widely used in industries like chemicals, foods, pharmaceuticals, bioprocessing, agrochemicals, biology, biomedicine, agriculture and medicine. One of the major objectives of these designs is to study the functional relationship between one or more responses and a number of quantitative input factors. However, biological materials have more run to run variation than in many other experiments, leading to the conclusion that smaller response surface designs are inappropriate. Thus designs to be used in these research areas should have greater replication. Gilmour (2006) introduced a wide class of designs called “subset designs” which are useful in situations in which run to run variation is high. These designs allow the experimenter to fit the second order response surface model. However, there are situations in which the second order model representation proves to be inadequate and unrealistic due to the presence of lack of fit caused by third or higher order terms in the true response surface model. In such situations it becomes necessary for the experimenter to estimate these higher order terms. In this study, the properties of subset designs, in the context of the third order response surface model, are explored.     

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.     

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.     

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.     

Some series of proper generalized neighbor designs

Neighbor balanced designs are used to remove the neighbor effects but most of these designs require a large number of blocks. To neutralize the neighbor effects in such situations, GN₂-designs are most desirable. This article deals with the (i) refinement of some series of GN₂-designs constructed by Zafaryab et al. (2010) and (ii) construction of some new series of GN₂-designs in circular blocks of equal size.     

Construction of repeated measurements designs strongly balanced for residual effects

Abstract

Repeated Measurements Designs have been widely used in agriculture, animal husbandry, education, biology, botany and engineering. Balanced or strongly balanced repeated measurements designs are useful to balance out the residual effects. In this article, some new generators and construction procedures are proposed to obtain circular strongly balanced repeated measurements designs in periods of (a) equal sizes, (b) two different sizes, and (c) three different sizes.