Neighbour Balanced Research Articles

Some New Dimensions to Construct Economical Circular Weakly Balanced Neighbor Robust Designs

Neighbor-balanced designs are used in the experiments where the performance of a treatment is affected by the treatments applied to its neighboring units. These are well-known designs to balance the neighbor effects. Among these designs, minimal neighbor-balanced designs are economical; therefore, these are preferred by the experimenters. For v even, minimal neighbor balanced designs in circular blocks cannot be constructed for most of the cases, where v is number of treatments. In such situations, experimenters would like to relax the neighbor balance property up to some extent and consider the minimal circular weakly balanced neighbor designs as the better alternates to the minimal circular neighbor-balanced designs. In this article, some generators are developed to obtain minimal circular weakly balanced neighbor designs in blocks of equal, two and three different sizes.

A new dimension of balanced universally one sided design and its satisfying optimal neighbour condition

In an agricultural design of experiment the treatment received by a plot may affect the other response on the neighbouring plots of a same block or it may happen to affect the response on the following plot. For example of the second condition, the tall varieties may affect the other crops grown on the neighbouring plots by their shades. Bailey (2003) has developed such design concerned with the study of one sided neighbour effect, under the above mentioned second condition. This paper gives a new series of Universally Optimal One-Sided Circular Neighbour Balanced designs.

Some useful classes of minimal weakly balanced neighbor designs in circular blocks of two different sizes

Neighbor designs (NDs) are used to balance out the neighbor effects. Among these designs, minimal NDs are economical, therefore, these are preferred by experimenters. Unfortunately, minimal NDs cannot be constructed for all combinations of v (number of treatments) and k (block sizes). Minimal weakly balanced NDs (WBNDs) are recommended in the situations where minimal NDs could not be constructed. In this article, some generators are developed to obtain minimal circular WBNDs (MCWBNDs) in blocks of two different sizes. In our proposed designs, 3v/2 unordered pairs of distinct treatments appear twice as neighbors while all other pairs appear once. In these designs, we lose % neighbor balance while saving at least % experimental material.

Generating row–column field experimental designs with good neighbour balance and even distribution of treatment replications

AbstractBlocking is a standard strategy for improving the precision of randomized experimental designs. In field trials, it is usually beneficial to impose blocks in both row and column directions. A common problem with classical randomized row–column designs is that replications of a treatment may be clustered and that some pairs of treatments appear next to each other rather more often than others. Such patterns adversely affect certain design properties which we refer to as neighbour balance (NB) and evenness of distribution (ED), and they may be particularly harmful if some of the treatments are very susceptible to environmental stresses. Here, we consider an experimental design strategy for blocked experiments that achieves good NB&ED properties. The approach is exemplified using five examples, spanning non‐resolvable and resolvable replicated designs as well as partially replicated designs.

Infinite Latin Squares: Neighbor Balance and Orthogonality

Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table. Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.

Error variance bias in neighbour balance and evenness of distribution designs

SummaryNeighbour balance and evenness of distribution designs help to address user concerns in the two‐dimensional layout of agricultural field trials. This is done by minimising the occurrence of pairwise treatment plot neighbours and ensuring that the replications of treatments are spread out across rows and columns of a trial. Such considerations result in a restriction on the normal randomisation process for a row‐column design which can lead to error variance bias. In this paper, uniformity trial data is used to assess the degree of this bias for both resolvable and non‐resolvable designs. Comparisons are made with a similar investigation using Linear Variance spatial designs.

Efficiency of Neighbouring Designs for First Order Correlated Models

The comparison of efficiency of Complete and Incomplete Nearest Neighbour Balanced Block Designs over regular block design using average variance, generalized variance and min-max variance with the error term e given in the NNBD model follows using first order correlated models. It is observed that, RH and RD show increasing efficiency values for direct and neighbour effects (left and right) for MA(1) models. The RA and RG show neither increasing nor decreasing efficiency values are observed for direct and neighbouring effects for AR(1) and MA(1) models. In the case of ARMA(1,1) model, neither increasing nor decreasing efficiency values have been observed for average variance and generalized variance. The RE shows decreasing efficiency values with p in the interval 0.1 to 0.4 for direct and neighbouring effects for AR(1), MA(1) and ARMA(1,1) models.

Neighbor balance and evenness of distribution of treatment replications in row-column designs.

Row-column designs allow error control in field experiments by blocking in two dimensions. While this strategy can capture spatial heterogeneity aligned with blocks and account for effects due to the farming operations along rows and columns, it suffers from the occasional clustered occurrence of several replications of the same treatment. This property of classical row-column designs has hampered their more widespread use in practice. A further issue of practical importance is the degree of neighbor balance of a design, that is, the frequency of adjacencies of pairs of treatments. This paper proposes two design strategies to simultaneously optimize the evenness of spatial distribution of treatment replication as well as neighbor balance. Three examples are given to illustrate the proposed methods and demonstrate that both approaches yield comparable and satisfactory results.

Development of some useful generators to obtain partially neighbor balanced designs

Neighbor balanced designs are robust to neighbor effects, therefore, these designs are used to balance out the neighbor effects. If a large number of experimental material is required for combinatorial neighbor balance then partially neighbor balanced designs should be recommended. In this study, some useful generators are developed to obtain the partially neighbor balanced designs in linear blocks of sizes 3–7.

Efficiency of Nearest Neighbour Balanced Block Designs for Second Order Correlated Error Structure

Efficiency of Nearest Neighbour Balanced Block Designs for Second Order Correlated Error Structure

Web generation of experimental designs balanced for indirect effects of treatments (WEB-DBIE)

Indirect effects are effects that occur in an experiment due to the units which are adjacent (spatially or temporally) to the unit being observed. Spatial indirect effects arise due to the treatments applied to the adjacent neighbouring units/plots and the designs so developed are called Neighbour Balanced Designs (NBDs). Whereas temporal indirect effects occur because of the carryover or residual effects in the periods following the periods of their direct application and the designs considering temporal effects are called Crossover Designs. A large number of such designs have been developed in the literature. For ready referencing and potential use of these designs, online software for generation of randomized layout of these designs is highly desirable. This paper describes the development of a web solution for generation of NBDs and Crossover Designs using client–server architecture along with an online catalogue of the designs within a permissible range. Designs generated through this software have wide applications in agricultural experiments, forestry and agro-forestry trials, polycross nurseries, serological experiments, animal nutritional trials, sensory evaluations, clinical trials, etc. WEB-DBIE is accessible any time from arbitrary platforms through internet. This software provides freely available solution for the researchers and students working in this area.

MV- Optimality of Nearest Neighbour Balanced Block Designs using Second order correlated Models for Five Treatments.

I. Introduction Serology is a branch of Biometrics which is concerned with the study of virus and viral preparations. Many studies concerned with viral preparation require the arrangement of antigens in a plate so that each antigen has two other antigens as its neighbours. In analysis of such experiment the classical statistical design may not perform efficiently. Therefore REES (1967) introduced neighbouring structure. The following is the experiment considered by REES (1967) for the use of Nearest Neighbour Balanced Block Designs. If the observations available are correlated, the usual assumptions like independence of observations in the analysis of classical comparative experiment may not perform valid. Therefore there is a necessity for the use of NNBD. In biometrical sciences we can cite many areas where this kind of correlated structure exists. Now consider the viral preparations. Let there be �� kinds of antigens to be arranged on b plates, each containing k antigens. Each antigen appears r times (but not necessarily on r different plates) and is a neighbour of every other antigen exactly λ times. Rees used circular neighbouring block design and he was used incomplete neighbor design (k<�� ) in his experiment.

Efficiency of NNBD over NNBIBD using First Order Correlated Models

Neighbour Balanced Block Designs, permitting the estimation of direct and neighbour effects, are used when the treatment applied to one experimental plot may affect the response on neighbouring plots as well as the response on the plot to which it is applied. The allocation of treatments in these designs is such that every treatment occurs equally often with every other treatment as neighbours. Neighbour Balanced Block Designs for observations correlated within a block have been investigated for the estimation of direct as well as left and right neighbour effects of treatments. It is observed that efficiency for direct as well as neighbour effects is high, in case of Complete block designs i.e.,

Non-binary neighbor balance circular designs for v=2 n and λ=2

Neighbor balance designs were first introduced by Rees (1967) in circular blocks for the use in serological research. Subsequently several researchers have defined the neighbor designs in different ways. In this paper, neighbor balance circular designs for ( k≤ v) block size are constructed for even number of treatments i.e. v=2 n. No such series of designs is known in literature. Two theorems are developed for circular designs. Theorem 1 gives the non-binary circular blocks, whereas Theorem 2 generates binary circular blocks when n≤4 and non-binary blocks for n>4. In suggested designs no treatment is ever a neighbor of itself. Blocks are constructed in such a way that each treatment is a right and left neighbor of every other treatment for a fixed number of times say λ. Sizes of initial circular blocks are not same. One main guiding principle for such designs is to ensure economy in material use.

Properties of Designs in Experiments on Spatial Lattices

ABSTRACT When spatial variation is present in experiments, it is clearly sensible to use designs with favorable properties under both generalized and ordinary least squares. This will make the statistical analysis more robust to misspecification of the spatial model than would be the case if designs were based solely on generalized least squares. In this article, treatment information is introduced as a way of studying the ordinary least squares properties of designs. The treatment information is separated into orthogonal frequency or polynomial components which are assumed to be independent under the spatial model. The well-known trend-resistant designs are those with no treatment information at the very low order frequency or polynomial components which tend to have the higher variances under the spatial model. Ideally, designs would be chosen with all the treatment information distributed at the higher-order components. However, the results in this article show that there are limits on how much trend resistance can be achieved as there are many constraints on the treatment information. In addition, appropriately chosen Williams squares designs are shown to have favorable properties under both ordinary and generalized least squares. At all times, the ordinary least squares properties of the designs are balanced against the generalized least squares objectives of optimizing neighbor balance.

Balanced colourings of strongly regular graphs

A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For example, colourings of the triangular graph give balanced designs for experiments where the experimental units are unordered pairs of people. An analogue of Fisher's inequality is proved. In this context, strongly regular graphs may be generalized to arbitrary association schemes. When the association scheme is a collection of circular blocks then a colouring is balanced if the design in blocks is a 2-design and there is undirectional neighbour balance at all distances around the circle.

Theory &amp; Methods: 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.

Randomization of Neighbour Balanced Designs

This paper is a simulation study on the influence of interference between treatments in field trials. The considerations in the paper are based on a simple model which includes additive neighbour effects of treatments. We use uniformity data where neighbour effects are added, to demonstrate the influence that these effects have on the validity of comparisons between treatments. The simulations illustrate that the influence of the neighbour effects is reduced if a neighbour balanced or a partially neighbour balanced design is used.

Efficient schemes for nearest neighbor load balancing

We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diffusion type. Within this framework we develop a new Optimal Polynomial Scheme (OPS) which we show to terminate within a finite number m of steps, where m only depends on the graph and not on the initial load distribution.We show that all existing diffusion load balancing algorithms, including OPS, determine a flow of load on the edges of the graph which is uniquely defined, independent of the method and minimal in the l2-norm. This result can also be extended to edge weighted graphs.The l2-minimality is achieved only if a diffusion algorithm is used as preprocessing and the real movement of load is performed in a second step. Thus, it is advisable to split the balancing process into the two steps of first determining a balancing flow and afterwards moving the load. We introduce the problem of scheduling a flow and present some first results on its complexity and the approximation quality of local greedy heuristics.

On the Efficiency of Nearest Neighbour Balanced Block Designs with Correlated Error Structure

AbstractNearest Neighbour Balanced Block Designs (NNBD) are widely used in biological and industrial experiments. In this article we examine the optimality and efficiencies of NNBD under various correlated error structure models. A numerical comparison of NNBD with regular block design (RBD) is also presented.