Minimal Neighbor Designs in Linear Blocks

Neighbor designs are useful to neutralize the neighbor effects. In literature, most of the constructed neighbor designs are in circular blocks but linear blocks have more practical application in field experiments. In this article, some infinite series of minimal neighbor designs are constructed in proper linear blocks. There are many situations where minimal neighbor designs cannot be constructed in proper linear blocks. To overcome this problem neighbor designs in improper linear blocks and GN₂-designs in proper linear blocks are constructed.     

Some Classes of Binary Block Neighbor Designs

Neighbor designs are useful to neutralize the neighbor effects. In this paper two classes of these designs are constructed in circular binary blocks of size 4, 8, …, 24. First class consists of six infinite series of nearest neighbor designs in which each pair of distinct treatments appears once as neighbors. Second class also deals with six infinite series of these designs in which each pair of distinct treatments appears twice as neighbors. A catalog of nearest neighbor designs is also compiled in circular binary blocks for odd number of treatments from 23 to 99.     

Construction of Neighbor-Balanced Designs in Linear Blocks

Neighbor-balanced designs are useful to remove the neighbor effects in experiments where the performance of a treatment is affected by the treatments applied to its adjacent neighbors. In this article, neighbor-balanced designs are constructed in linear blocks of (i) equal sizes and (ii) two different sizes k ₁ and k ₂.     

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.     

Proper generalized neighbor designs in circular blocks

Generalized neighbor designs relax the condition of constancy on the number of times two treatments occur as neighbors in a design. In this paper, we constructed generalized neighbor designs in circular blocks of equal size. Catalogs of these designs for block size 5, 7 and 9 are also compiled. All the designs given here are binary.     

A note on circular neighbor balanced designs

ABSTRACT

This paper describes some methods of constructing circular neighbor balanced and circular partially neighbor balanced block designs for estimation of direct and neighbor effects of the treatments. A class of circular neighbor balanced block designs with unequal block sizes is also proposed.     

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.     

Generalized neighbor designs with block size 3

A generalized neighbor design relaxes the equality condition on the number of times two treatments as neighbors in the design. In this article, we have considered the construction of some classes of generalized neighbor designs with block size k=3 by using the method of cyclic shifts. The distinguishing feature of this construction method is that the properties of a design can easily be obtained from the sets of shifts instead of constructing the actual blocks of the design. A catalog of generalized neighbor designs with block size k=3 is compiled for v∈{5,6,…,18} treatments and for different replications. We provide the reader with a simpler method of construction, and in general the catalog that gives an open choice to the experimenter for selecting any class of neighbor designs.     

Universally Optimal Designs Balanced for Neighbor Effects

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 this paper, criterion for construction of universally optimal neighbor balanced designs is discussed.     

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.     

Computer-Generated Neighbor Designs

Neighbor designs are useful to remove the neighbor effects. In this article, an algorithm is developed and is coded in Visual C + +to generate the initial block for possible first, second,…, and all order neighbor designs. To get the required design, a block (0, 1, 2,…, k ? 1) is then augmented with (v ? 1) blocks obtained by developing the initial block cyclically mod (v ? 1).     

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 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.     

Circular Binary Block Second- and Higher-Order Neighbor Designs

Some new neighbor designs are presented here. Second-order neighbor designs for different configurations are generated in circular binary blocks. Third-order and fourth-order neighbor designs for some cases are also constructed. In all cases, circular blocks are well separated and these designs are obtained through initial block/s. At the end of the study, some models for analysis of these designs are also presented.     

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.     

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.     

Some important classes of generalized neighbor designs for linear blocks

In this paper, some infinite series of generalized neighbor designs are constructed for the linear blocks which are useful to balance out the neighbor effects for the cases where (a) one of the v treatments has some neighbor effects with other treatments, while remaining (v – 1) treatments have half of that neighbor effect among selves, (b) some of the v treatments have some neighbor effect with other treatments, while remaining treatments have half of that neighbor effect among themselves, (c) one of the v treatments has some neighbor effect with other treatments, while remaining (v – 1) treatments have double of that effect among themselves, and (d) some of the v treatments have some neighbor effect with other, while remaining treatments have double of it among themselves.     

Minimal circular strongly balanced repeated measurements designs in periods of three different sizes

Repeated measurements designs (RMD) are widely used in medicine, pharmacology, animal sciences, and psychology. If there is a restriction on the total number of treatments, some experimental units can receive on the total length of time while some experimental units can remain in the trial, then RMD in periods of unequal sizes should be used. In this article, some infinite series are developed to generate the minimal circular strongly balanced RMD in periods of three different sizes p₁, p₂, and p₃, where 2 ≤ p₃ < p₂ ≤ 10.     

Circular balanced repeated measurement designs in periods of three different sizes

Abstract

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. If there is a restriction on the total number of treatments, some experimental units can receive on the total length of time while some experimental units can remain in the trial, then repeated measurements designs with unequal period sizes should be used. In this article, some infinite series are developed to generate the minimal balanced repeated measurement designs in periods of three different sizes p₁, p₂ and p₃, where 2?≤?p₃?<?p₂ ≤ 10 and p₂?<?p₁.     

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.