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.     

Constructions of Nested Pairwise Efficiency and Variance Balanced Designs

ABSTRACT

Nested pairwise efficiency and variance balanced designs form a new class of block designs. In this article, two methods of constructing such designs from a symmetric balanced incomplete block design are proposed with some illustrations.     

Construction of neighbor designs of block size 4 for all values of v ≥ 5 by using the method of addition

In this article, a general method of construction of neighbor block designs is given. The designs are constructed using variation of a simple method which we refer to as the method of addition (renamed as the method of cyclic shifts). We give complete solution of neighbor balanced designs for k = 4 for any value of v. We also give many series of generalized neighbor designs (GNDs). In the last section, we have constructed GNDs in a sequential manner (as Did John 1981) for v ≤ 50 and r is multiple of k.     

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.     

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.     

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 CONSTRUCTION OF GENERALIZED GROUP DIVISIBLE DESIGNS WITH TWO GROUPS

ABSTRACT

Many simple and general methods of construction, using α-resolvable balanced incomplete block designs, for generalized group divisible designs with two groups (GGDD(2)) are given. The E-optimality of all constructed designs is also discussed.     

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.     

Designs balanced for neighbor effects in circular blocks of size six

The performance of a treatment is affected by the treatments applied to its adjacent plots, especially in the experiments of agriculture, horticulture, forestry, serology and industry. Neighbor designs ensure that treatment comparisons are least affected by neighbor effects, therefore, this is a rich field of investigation in statistics and in combinatorics. In this article, several series of neighbor balanced designs are considered in circular blocks of six units.     

On Universal Optimality of Circular Weakly Neighbor Balanced Designs Under an Interference Model

In this article, the optimal design problem in a fixed effects interference model with left-neighbor effects is studied. It is known (Druilhet, 1999 Druilhet , P. ( 1999 ). Optimality of circular neighbor balanced designs . J. Statist. Plann. Infer. 81 : 141 – 152 .[Crossref], [Web of Science ®] , [Google Scholar]) that circular neighbor balanced designs (CNBDs) are universally optimal in such a model. We prove the universal optimality of circular weakly neighbor balanced designs (CWNBDs), which require a smaller number of blocks than CNBDs. CWNBDs with the number of blocks smaller than the number of treatments belong to the class of partially neighbor balanced designs (PNBDs) defined by Wilkinson et al. (1983 Wilkinson , G. N. , Eckert , R. , Hancock , T. W. , Mayo , O. ( 1983 ). Nearest neighbour (NN) analysis of field experiments . J. Roy. Statist. Soc. Ser. B 45 : 151 – 210 . [Google Scholar]). We give a construction method for some CWNBDs, with examples.     

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

Families of proper generalized neighbor designs

A generalized neighbor design relaxes the equality condition on the number of times two treatments occur as neighbors in the design. In this article we have constructed a new series of generalized neighbor designs with equal block sizes, a series of neighbor designs of Rees [1967. Some designs of use in serology. Biometrics 23, 779–791] and a series of neighbor designs with two distinct block sizes. Two more new series of GN₂ designs are also constructed for even number of treatments. It has been shown that quasi neighbor designs introduced by Preece [1994. Balanced Ouchterlony neighbor designs. J. Combin. Math. Combin. Comput. 15, 197–219] are special cases of generalized neighbor designs with t=2$t=2$. All the designs given here are binary. A new definition—partially balanced circuit design is introduced which is a special case of generalized neighbor designs with binary blocks.     

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.     

Three-factor symmetrical balanced designs with full efficiency for main effects

Some methods for constructing balanced design for 3-factor symmetrical factorial experiments in which all the main effects are completely unconfounded by using balanced arrays and BIB designs are proposed. The method is flexible in terms of selecting block size.     

An infinite class of PBIB designs

In this paper, an infinite class of partially balanced incomplete block (PBIB) designs of m+1 associate classes is constructed through the use of a series of row-orthogonal matrices known as partially balanced orthogonal designs (PBOD) of m-associate classes. For the purpose, a series of PBOD is obtained through a method described herein. An infinite class of regular GD designs is also reported.     

On the E-optimality of complete block designs under a mixed interference model

In experiments in which the response to a treatment can be affected by other treatments, the interference model with neighbor effects is usually used. It is known that circular neighbor balanced designs (CNBDs) are universally optimal under such a model if the neighbor effects are fixed (Druilhet, 1999) or random (4 and 7). However, such designs cannot exist for every combination of design parameters. In the class of block designs with the same number of treatments as experimental units per block, a CNBD cannot exist if the number of blocks, b  , is equal to p(t−1)±1$p(t-1)\pm 1$, where p is a positive integer and t is the number of treatments. Filipiak et al. (2008) gave the structure of the left-neighboring matrix of E-optimal complete block designs with p  =1 under the model with fixed neighbor effects. The purpose of this paper is to generalize E-optimality results for designs with p∈N$p\in \mathbb{N}$ assuming random neighbor effects.     

Universally optimal designs under interference models with and without block effects

The concept of neighbor designs was introduced and defined by Rees (1967 Rees, D.H. (1967). Some designs of use in serology. Biometrics 23:779–791.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) along with giving some methods of their construction. Henceforth, many methods of construction of neighbor designs as well as of their generalizations are available in the literature. However, there are only few results on their optimality. Therefore, the purpose of this article is to give an overview of study on this problem. Recent results on optimality of specified neighbor balanced designs under various interference models with block effects are presented and then these results are compared with respective models where block effects are not significant.     

Neighbor-Balanced Row-column Designs

In this article, row-column designs incorporating directional neighbor effects have been studied. A row-column design is said to be neighbor balanced if every treatment has all other treatments appearing as a neighbor a constant number of times. We considered here three different situations under row-column setup incorporating neighbor effects viz., row-column design with one-sided neighbor effect, two-sided neighbor effect, and four-sided neighbor effect. The information matrices for all the situations for estimating the direct and neighbor effects of treatments have been derived. Methods of constructing neighbor-balanced row-column designs have been developed and its characterization properties have been studied.     

On the Optimality of Circular Block Designs Under a Mixed Interference Model

Filipiak and Markiewicz (2012 Filipiak, K., Markiewicz, A. (2012). On universal optimality of circular weakly neighbor balanced designs under an interference model. Comm. Stat. Theor Methods 41: 2356–2366.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) proved the universal optimality of circular weakly neighbor balanced designs (CWNBDs) under the interference model with fixed neighbor effects among the class of complete block designs. In two special cases where a CWNBD cannot exist, Filipiak et al. (2012 Filipiak, K., Markiewicz, A., Ró?ański, R. (2012). Maximal determinant over a certain class of matrices and its application to D-optimality of designs. Linear Algebra Appl. 436(4): 874–887.[Crossref], [Web of Science ®] , [Google Scholar]) characterized D-optimal designs. The aim of this paper is to show the universal optimality of CWNBDs and to characterize D-optimal designs under the interference model with random neighbor effects.