Science in China Series A: MathematicsOct., 2009, Vol. 52, No. 10, 2243–2256www.scichina.com math.scichina.comwww.springerlink.com

Circular neighbor-balanced designs using cyclicshifts

IQBAL Ijaz1†, TAHIR M. H.2 & GHAZALI Syed Shakir Ali2

1 Department of Statistics, Bahauddin Zakariya University, Multan 60800, Pakistan2 Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

(email: drijaziqbal@yahoo.com, mtahir stat@yahoo.com, shakirghazali@yahoo.com)

Abstract In agriculture experiments, the response on a given plot may be affected by the treatments

on neighboring plots as well as by the treatments applied to that plot. In this paper we consider such

type of situations and construct circular neighbor-balanced designs (CNBDs) by the method of cyclic

shifts or sets of shifts. An important feature of this method is that the properties of a design can be

easily obtained from the sets of shifts instead of constructing the actual blocks of the design. That

is, the off-diagonal elements of the concurrence matrix can be easily obtained from the sets of shifts.

Since the suggested designs are circular, balanced and binary, so they are universally optimal.

Keywords: A-optimal, circular designs, cyclic shifts, treatment-balanced designs, universally

optimal

MSC(2000): 05B05, 62K05, 62K10

1 Introduction

In some countries, the agricultural land is found to be on terraces cut into hillsides. Theseterraces often encircle a hill and so naturally form circular blocks. If experiments are undertakenby using such blocks, it might be appropriate to allow for the effects of neighboring treatmentsbecause in such cases, the response on a given plot may be affected by the treatments appliedto that plot as well as by the treatments on neighboring plots[1]. These neighboring effectsare sometimes known as competition effects or interference effects. If these circular neighbordesigns are balanced, then it may prove helpfully in removing unnecessary (residual) effects ofneighbors.

In this paper, we consider the construction of circular neighbor-balanced designs (CNBDs)and construct several series of such designs by the method of cyclic shifts. These suggesteddesigns are circular, balanced, binary and optimal. In some series, the proposed designs areneighbor-balanced at distances up to k − 1, where k is the block size. Some definitions andterminologies are given below.

Definition 1.1[1]. A block design is balanced if each pair of treatments occurs in the samenumber of times.

Definition 1.2. A block design is a balanced incomplete block design (BIBD) for v treatments

Received June 23, 2008; accepted February 12, 2009DOI: 10.1007/s11425-009-0063-1† Corresponding author

Citation: Iqbal I, Tahir M H, Ghazali S S A. Circular neighbor-balanced designs using cyclic shifts. Sci China SerA, 2009, 52(10): 2243–2256, DOI: 10.1007/s11425-009-0063-1

2244 Iqbal I et al.

in b blocks each of size k (k < v) and r replications, if each pair of treatments occur same numberof times within blocks, say λ.

A block of k treatments {0, 1, . . . , v − 1} arranged in the circular order 0 → 1 → · · · →(v − 1) → 0 → · · · is said to circularly ordered. Denote by Ω(v,b,k) the set of all block designswith v treatments and b blocks of size k in which the k treatments in each block are circularlyordered. Assuming in this paper that k � v, and l = bk[v(v − 1)]−1 and λ = l(k − 1) areintegers, we only consider circular binary block designs.

Definition 1.3[1]. A circular binary block design is a circular block design in which eachtreatment occurs at most once in each block.

Definition 1.4. If each treatment has every other treatment as neighbor equal number oftimes, then the design is said to be a CNBD.

Definition 1.5[1]. A design is a circular neighbor-balanced design at distance i � k − 1 ifit is circular binary block design in Ω(v,b,k), and is a BIBD such that for each unordered pairof distinct treatments, there exists exactly l plots such that each of these plots receives the firstchosen treatments and the right neighbor at distance i receives the second treatment. A circularblock design is said to be neighbor-balanced at distances up to γ, denoted by CNBD(v, b, k; γ),if it is neighbor-balanced at distance i for all 1 � i � γ.

The following example shows a CNBD using Definition 1.5.

Example 1.1. A CNBD(5, 4, 5; 4) can be constructed by using the following sets of shifts[1, 1, 1, 1]15 + [2, 2, 2, 2]15 + [3, 3, 3, 3]15 + [4, 4, 4, 4]15 .

0 0 0 0

1 2 3 4

2 4 1 3

3 1 4 2

4 3 2 1

The above design is neighbor-balanced at distances up to k − 1. Further detail on constructionof CNBDs is given in Section 4.

Now, we discuss the role of concurrence matrix in balanced block design (or in BIBD).

In a BIBD, (v, k, r, b, λ) is the set of parameters and has the relation vr = bk and λ =r(k − 1)/(v − 1). Let N = (nij) be the (v × b) treatment-block incidence matrix associatedwith any block design d ∈ D (v, b, k) whose elements nij signify the number of units in blockj allocated to treatment i. The matrix NN ′ is referred to concurrence matrix of design d, andits entries are denoted by λij . For any block design, NN ′, the treatment concurrence withdiagonal elements equal to r and off-diagonal elements are equal to the number of times anypair of treatment occur together within blocks. In a BIBD, the diagonal elements are equal tor and all off-diagonal entries of NN ′ are equal to λ.

The information matrix for estimating treatment contrasts is

C = rδ − k−δNN ′ ,

Circular neighbor-balanced designs using cyclic shifts 2245

where kδ is the matrix with diagonal elements equal to the corresponding elements of k andoff-diagonal elements equal to zero. C is symmetric and non-negative definite, with rank v − 1for d ∈ D. C-matrix determines the statistical properties of a BIBD and also plays a key rolein establishing optimality criteria.

The concurrence matrix, NN ′, and the concurrences among off-diagonal elements play a vitalrole for a block design to be A-optimal or not. According to [2], the information matrix C andthe concurrence matrix NN ′ of a cyclic design are circulant. Furthermore, the circulant matrixcan be specified by the elements in the first row, since the other rows are obtained from the firstrow by a cyclic rotation. Due to this reason, we can quickly and easily obtain the properties ofa CNBD directly from the shifts without constructing blocks of the design.

Rees[3] first introduced the concept and the notion of neighbor designs and stated that aneighbor design (or a circular block design) for v treatments is an arrangement of b blocksof size k, such that each treatment appears r times and any two distinct treatments appearas neighbors, that is, with adjacent positions, λ1 times. The parameters of neighbor designsare v, k, r, b and λ1, and these parameters are not independent and satisfy the relationsvr = bk, λ1(v−1) = 2r. The subsequent work on the construction of neighbor designs appearedin [4–16] and the optimality issue of neighbor-balanced designs has been discussed in [1, 17–19].

Rees[3] did not consider the situation where an observation would be taken on each plot ineach block. In experimental situations considered by Rees no linear model for these observationswas necessary or appropriate. We introduce the idea of using linear model for the constructionof CNBDs, which also provides treatment-balanced designs. This model may be considered asa model for data observed in circular blocks or as a tool for understanding the properties ofCNBDs.

In this article, we give a general method of construction of CNBDs, which is based on thecyclical developments of one or more sets of v plots. We refer to this cyclical development asthe method of cyclic shifts or sets of shifts, which is also used by Iqbal and Tahir[20,21] andIqbal et al.[22].

This article is organized as follows: The model for CNBDs is given in Section 2. The methodof cyclic shifts is described in Section 3. The construction of CNBDs is given in Section 4. Andin Section 5 we conclude this paper with some remarks.

2 The model

Consider a design for v treatments in b blocks of size k. We assume the following model for Yij

observations, and let Yij be the response on plot i in block j, then the model will be

Yij = μ + τi + β(i+1) + β(i−1) + αj + εij , i = 1, 2, . . . , k, j = 1, 2, . . . , b, (1)

where μ is the general mean, τi is the direct effect of the treatment applied to plot i, β(i+1) isthe right-neighbor effect of the treatment on plot (i + 1)mod (k + 1), β(i−1) is the left-neighboreffect of the treatment on plot (i − 1)mod (k + 1), αj is the effect of block j and εij is anindependent random error with mean zero and variance σ2.

In matrix form, the model will be

Y = μX0 + X1τ + X2β + X3α + ε, (2)

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where Y is (bk × 1) vector of response, X0 is the (bk × 1) vector of 1s, X1 is the (bk × v)incidence matrix for treatment effects, X2 is the (bk × v) incidence matrix for neighbor effects,X3 is the (bk × b) incidence matrix for block effects, ε is the (bk × 1) vector of random errors,τ is the (v × 1) vector of treatment effects, β is the (v × 1) vector of neighbor effects, and α isthe (b × 1) vector of block effects.

LettingX = [X0 : X1 : X2 : X3] and π′ = [μ′, τ ′, β′, α′],

the model reduces to Y = Xπ + ε.

The set of normal equations required to obtain the least squares estimate of π̂ is

(X ′X)π̂ = X ′Y,

where

X ′X =

⎡⎢⎢⎢⎢⎢⎣

X ′0X0 X ′

0X1 X ′0X2 X ′

0X3

X ′1X0 X ′

1X1 X ′1X2 X ′

1X3

X ′2X0 X ′

2X1 X ′2X2 X ′

2X3

X ′3X0 X ′

3X1 X ′3X2 X ′

3X3

⎤⎥⎥⎥⎥⎥⎦

, X ′Y =

⎡⎢⎢⎢⎢⎢⎣

G

T

R

B

⎤⎥⎥⎥⎥⎥⎦

and π̂ =

⎡⎢⎢⎢⎢⎢⎣

μ̂

τ̂

β̂

α̂

⎤⎥⎥⎥⎥⎥⎦

.

Here G = X ′0Y , T = X ′

1Y , R = X ′2Y and B = X ′

3Y .Let L = X ′

1X2 and M = X ′2X2.

We also note that

X ′0X0 = bk = rv, X ′

0X1 = rE1,v, X ′0X2 = 2rE1,v, X ′

0X3 = kE1,b, X ′1X1 = rIv ,

X ′1X3 = N, X ′

2X3 = 2N and X ′3X3 = kIb,

where E1,v is a matrix of dimension (1 × v) containing all 1’s.Under the restriction E1,v τ̂ = E1,vβ̂ = 0, E1,bα̂ = 0, we have μ̂ = G/rv, and hence the

normal equations obtained are

rμ̂Ev,1 + rIv τ̂ + Lβ̂ + Nα̂ = T,

2rμ̂Ev,1 + L′τ̂ + Mβ̂ + 2N = R,

kμ̂Eb,1 + N ′τ̂ + 2N ′β̂ + k Ivα̂ = B.

After simplification, the adjusted least squares equations for treatment effects τ̂ and neighboreffects β̂ are

⎡⎣ r Iv − (1/k)NN ′ L − (2/k)NN ′

L′ − (2/k)NN ′ M − (4/k)NN ′

⎤⎦

⎡⎣ τ̂

β̂

⎤⎦ =

⎡⎣ T − (1/k)NB

R − (2/k)NB

⎤⎦ .

It will be noted that NN ′ is the treatment concurrence matrix whose (i, i′)-th element is thenumber of times treatments i and i′ occur together in the same block, i �= i′ and the (i, i′)-thelement of L is the number of times treatment i′ is a neighbor of treatment i, where i �= i′.That is, we cannot distinguish between right- and left-neighbors while constructing the matrixL. M is a matrix whose (i, i′)-th element is the number of times treatment i and i′ have acommon neighboring plot for i �= i′.

Circular neighbor-balanced designs using cyclic shifts 2247

Proposition 2.1. A CNBD is neighbor-balanced if all off-diagonal elements of the L matrixare equal.

Proposition 2.2. A CNBD is treatment-balanced design if all off-diagonal elements of theconcurrence matrix, NN ′, are equal. To obtain a treatment-balanced design, it is also necessarythat all other off-diagonal elements of M must be equal.

3 The method of cyclic shifts

The method of cyclic shifts is a particular way of constructing cyclic BIBDs or CNBDs. Herethe v treatments are labeled as 0, 1, 2, . . . , v−1 and we consider the construction of equireplicatebinary designs (a binary design in which each element of the incidence matrix, N , is either 0or 1) for v treatments in b = v blocks of size k. The method of construction is to allocate tothe first plot in the i-th block the treatment i, i = 0, 1, 2, . . . , v − 1. We denote this using thevector u1 = [0, 1, 2, . . . , v − 1], which holds the treatments allocated to the first plot in each ofthe blocks 1, 2, . . . , v respectively. To obtain the treatment allocation of the remaining plots ineach block, we cyclically shift the treatments allocated to the first plot. In order to define acyclic shift, let ui denote the allocation of treatments to the i-th plot in each block. That is, thej-th element of ui is the treatment allocated to plot i of block j. A cyclic shift of size qi, whenapplied to plot i, is then such that ui+1 = [ui + qi1], where addition is mod v, 1 is a vector ofones, 1 � i � k−1 and 1 � qi � v−1. Assuming that we always start with u1 as defined above,a design is completely defined by a set of k − 1 shifts, Q, say, where Q = [q1, q2, . . . , qk−1]. Toavoid a treatment occurring more than once in a block we must ensure that the sum of any twosuccessive shifts, the sum of any three successive shifts, . . ., the sum of any k − 1 successiveshifts, is not equal to zero mod v. Subject to this constraint, Q may consist of any combinationof shifts including repeats. Also, the shifts need only range from 1 to [v/2] inclusive, where[v/2] is the greatest integer than or equal to [v/2]. This is because a shift of size q is equivalentto one of size v − q (mod v).

To illustrate the above method of construction, let us consider the construction of design forv = 5 and k = 3. The possible shifts are defined as Q = [q1, q2], where qi= 1 or 2 and i= 1 or2. Two possible choices are Q1 = [1, 1], and Q2 = [1, 2]. Using Q1, we get u2 = [1, 2, 3, 4, 0]and u3 = [2, 3, 4, 0, 1]. The complete designs obtained from using Q1 and Q2 are given belowas Design 1 and Design 2. The blocks of designs are written vertically.

0 1 2 3 4

1 2 3 4 0

2 3 4 0 1

0 1 2 3 4

1 2 3 4 0

3 4 0 1 2

Design 1 Design 2

The properties of a design depend on the number of concurrences between the pair of treatments.A concurrence between two treatments occur when both treatments are in the same block. InDesign 1, for treatment 0 the concurrences with each of the treatments 1, 2, 3 and 4, respectivelyare 2, 1, 1, 2, while in Design 2 the corresponding concurrences are 1, 2, 2, 1.

Because of the cyclic nature of the construction, the number of concurrences between anytreatment and the remainder can be obtained from the number of concurrences between treat-ment 0 and the remainder. Also, the number of concurrences between treatment 0 and the

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remainder can be easily obtained from Q, the set of shifts used to construct the design.Similarly, the number of concurrences between treatment 1 and treatments 2, 3 and 4 are

2, 1 and 1 respectively, that is, they are obtained by cycling the list of concurrences. To showthat the number of concurrences between treatments can be obtained from Q, we consider thenumber of concurrences for Design 1 and the shifts Q1 = [1, 1] used to construct this design.We note that the number of concurrences are symmetric about [v/2] in the sense that any shiftof size q that results in a concurrence between treatment 0 and treatment q also results in aconcurrence between treatment 0 and treatment v − q. This means that we need to use onlyshifts such that 1 � qi � [v/2] when defining Q.

If shifts q1 and q2 are applied successively to treatment 0, the result is a concurrence betweentreatment 0 and treatments q1 and q2, and a concurrence between treatment 0 and treatmentq1 + q2. If a third shift q3, say, then is applied after q1 and q2, the following treatments will alsoconcur with treatment 0 : q1+q2+q3, q2+q3 and q3. This adding of shifts to get the treatmentswhich concur treatment 0 works for the general case and so enables the number of concurrencesof a design to be obtained directly from the shifts which define it. In the general case, if shiftsq1, q2, . . . , qi−1 are applied to treatment 0, then the additional concurrences, which result whenshift qi is applied, are between treatment 0 and treatments q1 + q2 + · · ·+ qi, q2 + q3 + · · ·+ qi,. . ., qi−1 + qi, qi, where addition is mod v. To illustrate the calculation of the number ofconcurrences from the shifts, consider the Design 1 again. The shifts are Q1 = [1, 1]. Thetreatments which concur with treatment 0 are therefore 1,1 and 1+1, i.e. treatment 1 concurstwice and treatment 2 concurs once. By the symmetry property, treatment 0 also concurs twicewith treatment 4 and once with treatment 3. The concurrences of treatment 1 with treatments2, 3 and 4 are 2, 1 and 1 respectively. Similarly, the concurrences of treatment 2 with 3 and 4are 2 and 1 respectively, and so on. For more details see [23].

3.1 Fractional designs

A feature of the method of construction is that certain sets of shifts produce design that aremade up of complete replications of the smaller designs. That is, the v blocks can be dividedinto s sets of size n = v/s and the n blocks in each set contain the same treatment allocation.For example consider v = 6, k = 4 and Q = [1, 2, 1]. The design obtained by applying Q to u1

is given below:

0 1 2 3 4 5

1 2 3 4 5 0

3 4 5 0 1 2

4 5 0 1 2 3

Design 3

It appears from the above design that it is made up from two complete designs as indicated bythe central vertical line. Each set contains three blocks of treatments (0, 1, 3, 4), (1, 2, 4, 5)and (0, 2, 3, 5). The number of concurrences that treatment 0 makes with other treatments arerespectively 2, 2, 4, 2, 2 in the whole design whereas the concurrences are 1, 1, 2, 1, 1 (half) inthe fractional design and is denoted by [1, 2, 1]12 .

In order to decide whether a set of shifts will produce a fractional design, we will take the

Circular neighbor-balanced designs using cyclic shifts 2249

value of v/k and determine the smallest integer z, say, which makes (v× z)/k an integer n, say.If v is also divisible by n then m fractional designs can be obtained.

Let v/n = m, then we have m fractional designs within the whole design with z the numberof replicates of each fractional design and n the number of blocks of each fractional design. Theshifts that will produce the required design must be such that the sum of every z successiveshifts is equal to n. In the above example, we have

v

k=

64,

v × z

k=

6 × 24

= 3 = n,v

n=

63

= 2 = m.

That is m = 2 and there are two fractional designs within the whole design. Each fractionaldesign has n (= 3) blocks, z (= 2) replications and the shifts used are Q = [1, 2, 1], that is, thesum of z (= 2) successive shifts is equal to n (= 3).

A CNBD can also be constructed by adding one or more subsets of blocks contained withinthe design to one or more other subsets or complete designs. In order to construct a CNBDwith more than v blocks, we combine the blocks obtained from more than one sets of shifts.As an illustration, given below is a design for v = 6 treatments in 15 blocks of size 4 which hasbeen constructed by combining together the blocks which are obtained from the three sets ofshifts [1, 1, 2], [1, 1, 3] and [2, 3, 4]12 .

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2

1 2 3 4 5 0 1 2 3 4 5 0 2 3 4

2 3 4 5 0 1 2 3 4 5 0 1 5 0 1

4 5 0 1 2 3 5 0 1 2 3 4 3 4 5

Design 4

The above design has been constructed by using shifts [1, 1, 2] + [1, 1, 3] + [2, 3, 4]12 , wherethe “+” signs indicate that the blocks constructed from the separate sets of shifts must becombined together.

3.2 Adding a new treatmentSometimes a design for v − 1 treatments with block of sizes k and k − 1 can be convertedinto a neighbor-balanced design for v treatments, and all blocks of size k by adding additionaltreatment to each of the smaller blocks of size k − 1. The example of such design for v = 8,b = 14 and λ1 = 2 is given below. The two sets of shifts used to construct the original designfor v = 7, k1 = 4 and k2 = 3 are Q1 = [1, 2, 3] and Q2 = [2, 4]. We note that treatment 7 hasbeen added to each block of the smaller design.

0 1 2 3 4 5 6 0 1 2 3 4 5 6

1 2 3 4 5 6 0 2 3 4 5 6 0 1

3 4 5 6 0 1 2 6 0 1 2 3 4 5

6 0 1 2 3 4 5 7 7 7 7 7 7 7

Design 5

4 Construction of CNBDs

In this section we will first explain how to find the neighbors of treatment 0. The cyclic methodof construction (or cyclic shifts) ensures that if we are able to find the neighbors of treatment0, then the neighbors of all the other treatments follow the same pattern.

2250 Iqbal I et al.

If treatment 0 is on plot i in block j, then the treatment q, say, on plot (i + 1) mod (k + 1)is the right-neighbor of treatment 0 in that block. The treatment on plot (i − 1) mod (k + 1)in block j is the left-neighbor of treatment 0 in block j. The cyclic method of constructionensures that if any treatment q, say, is a right-neighbor of treatment 0 in block j, then treatmentv− q mod (v) will be left-neighbor of treatment 0 in some other block j′, j′ �= j. This propertymakes the counting up of the right- and left-neighbors of treatment 0 quite easy. If a shift q,say, is applied to treatment 0, then treatment q will be a right-neighbor of treatment 0. Insome other block treatment v − q mod (v) will be a left-neighbor of treatment 0. To give it aname, we call v − q mod (v) the complement of shift q. Therefore, given a set of shift Q, wecan determine all the right- and left-neighbors of treatment 0 from the shifts contained in Q.The shifts q1, q2, . . . , qk−1 define the right-neighbors of treatment 0 and q1 + q2 + · · · + qk−1,mod (v) defines the left-neighbor of treatment 0 in the first block. By taking the complementsof these k shifts, we get all the remaining neighbors of treatment 0.Example 4.1. For v = 7, k = 3 and by using the set of shifts [1, 2], the design will be

0 1 2 3 4 5 6

1 2 3 4 5 6 0

3 4 5 6 0 1 2

Design 6

where the columns represent blocks. It can be seen from the above design that two of theright-neighbors of treatment 0 are the shifts 1 and 2, and the left-neighbors of treatment 0 inthe first block is treatment 3=1+2. The other neighbors of treatment 0 are the complement ofthese, that is, 6, 5 and 4. The above design is neighbor-balanced because treatment 0 has eachother treatment once as a neighbor. The cyclic method of construction ensures that if treatment0 has every other treatment as a neighbor an equal number of times, then each treatment i;i = 1, 2, . . . , v − 1 has each other treatment i′; i �= i′, as a neighbor an equal number of times.

The conditions that the shifts must satisfy to produce a neighbor-balanced design are givenin Theorem 1 below. The above design is also treatment-balanced, since at the same time eachtreatment pair has one neighboring plot in common, that is, the off-diagonal elements of thematrix M are equal. We will also explain below how we can get the off-diagonal elements of M

from the shifts.Now, we will consider an other example in which all the three matrices NN ′, M and L are

balanced.

Example 4.2. For v = 7, k = 4 and by using sets of shifts [1, 1, 1], [2, 2, 2] and [3, 3, 3], thedesign will be

0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

1 2 3 4 5 6 0 2 3 4 5 6 0 1 3 4 5 6 0 1 2

2 3 4 5 6 0 1 4 5 6 0 1 2 3 6 0 1 2 3 4 5

3 4 5 6 0 1 2 6 0 1 2 3 4 5 2 3 4 5 6 0 1

Design 7

The above Design 7 is treatment-balanced. Since the treatment 0 appears four times as aneighbor of each other treatment, so the above design is also neighbor-balanced. Furthermore,

Circular neighbor-balanced designs using cyclic shifts 2251

each treatment pair has a neighboring plot in common four times, so Design 6 also gives abalanced M matrix.

We will show that the treatments in the first row of each of the matrices NN ′, M and L

can be easily obtained from the shifts used to construct the design. As these three matrices arecirculant, they are completely defined by their first row.

To find how the concurrence matrix, NN ′, can be developed from the set(s) of shifts, thereaders may refer to Iqbal and Tahir[21] and Iqbal et al.[22].

To obtain the off-diagonal elements the matrix M , which give the number of times each pairof treatments has a neighboring plot in common, we construct the lists (i) and (ii) which aredefined as follows:

list (i)=[(q1 + q2), (q2 + q3), . . . , (qk−1 + qk)], andlist (ii)=[(q1 + q2 + · · · + qk−2), (q2 + q3 + · · · + qk−1), (q3 + q4 + · · · + qk)].Take the complement of each member of lists (i) and (ii). Then count the number of times

each of the treatment labels 1, 2, . . . , v − 1 occurs in the lists (i) and (ii), and in their comple-ments. These counts are the off-diagonal elements in the first row of M . If these lists containeach elements equal number of times then M matrix is balanced. The remaining rows can thenbe obtained by cyclically shifting the entries in the first row.

The calculation of the off-diagonal elements of L is described in Theorem 1 below.

Theorem 1. A neighbor-balanced for v treatments in b = (n × v) blocks of size k can beconstructed using n sets of shifts Q1, Q2, . . . , Qn if and only if the sets Q1, Q2, . . . , Qn andthe shifts

∑Q1,

∑Q2, . . . ,

∑Qn (where

∑Qi is the sum of all the shifts in list (i)), contain

between them all the possible shifts 1, 2, . . . , v − 1, an equal number of times, where both eachshift and its complement are counted.

Proof. Consider a particular set of shifts Qi. From the cyclic nature of construction, eachshift in the set identifies a treatment that will be a right-neighbor of treatment 0. The shifts∑

Qi identifies a treatment that will be left-neighbor of treatment 0. The remaining neighborsof treatment 0 are obtained from the complements of the shifts Qi and

∑Qi. Therefore, if the

complete collection of shifts, listed in the theorem, and their complements contain every shiftan equal number of times, then the design must be neighbor-balanced.

Similarly, we can construct CNBDs by using more than two sets of shifts. We can alsoconstruct CNBDs by combining fractional designs with other full or fractional designs.

In Theorem 1, designs were combined to give the final design. In this combining operationwe can also include subsets of the blocks to which we call fractional designs. The constructionof such designs is explained in Subsection 3.1.

Furthermore, the CNBDs can be obtained by adding an additional treatment. More detailswere given in Subsection 3.2. The conditions which must be satisfied in order that adding anadditional treatment leads to a CNBDs, are given in the following theorem.

Theorem 2. Suppose that D1 and D2 are two designs for v − 1 treatments such that in D1

there are b1 = n1v blocks of size k and the treatment replication is r1 and in D2 there areb2 = n2v blocks of size k−1 and the treatment replication is r2. Let Q1i (i = 1, 2, . . . , n1) be theset of shifts used to construct D1 and Q2j (j = 1, 2, . . . , n2) be the set of shifts used to constructD2. A neighbor-balanced design for v treatments in b1+b2 blocks of size k and replication r1+r2

2252 Iqbal I et al.

can be constructed by adding treatment v to each of the blocks in D2 and taking the completedesign as consisting of all the blocks in D1 and D2 if:

(i) Q1i (i = 1, 2, . . . , n1),∑

Q1i and Q2j (j = 1, 2, . . . , n2) contain every shift and its com-plement λ1 times where λ1 � 1,

(ii) b2 = r1 + r2, and(iii) the first plot in each block of D2 and the kth plot in each block of D2 contain between

them λ1 complete replicates of treatments 1, 2, . . . , v − 1.

Proof. Condition (i) ensures that without treatment v, designs D1 and D2 make up a neighbor-balanced design if in D1 we do not count the left-neighbor of the treatment on plot 1 in eachblock and in D2 we do not count the right-neighbor of the treatment on plot k in each block.Condition (ii) ensures that the quality of the treatment replication is maintained and condition(iii) ensures that when treatment v is added to D2 this treatment has every other treatment asa left- or right-neighbor λ1 times.

We note that it may be possible to obtain a neighbor-balance design for block size k byadding a treatment directly to a design for block size k− 1. That is, we do not necessarily haveto use design D1 in addition to D2.

Now consider the construction of CNBDs, which are treatment-balanced and many of theseare M -balanced because their off-diagonal elements of M -matrix are equal. All these types ofdesigns are optimal for the estimation of treatment and neighbor effects in the sense that theyare as balanced as it possibly be. For cases where we are unable to find a treatment-balanceddesign, the regular graph method is useful.

Now we will construct different series of CNBDs by using Theorem 1 and 2 as the followings:

Series 1. For k = 3, v is odd and v−13 is not a multiple of 3, the sets of shifts for possible

CNBDs are given as: [1, 1] + [2, 2] + · · · + [v−12 , v−1

2 ].

Series 2. For k = 3 and v is odd, the sets of shifts for possible CNBDs are given below:v = 7: [1, 2],v = 9: [1, 2]+[4(1

2 )]t,v = 13: [1, 3]+[2, 5],v = 15: [1, 3,]+[2, 6]+[5, 5]13 ,v = 19: [1, 4]+[2, 6]+[3, 7],v = 21: [1, 2]+[5, 6]+[4, 8]+[7, 7]13 ,v = 25: [1, 11]+[2, 7]+[3, 5]+[4, 6],v = 27: [1, 4]+[2, 10]+[3, 8]+[6, 7]+[9, 9]13 ,v = 31: [1, 14]+[2, 4]+[3, 5]+[7, 11]+[9, 10],v = 37: [1, 13]+[2, 3]+[4, 11]+[6, 10]+[7, 12]+[8, 9].

Series 3. For k = 3 and v is even, the sets of shifts for possible CNBDs are given below:v = 4: [1, 2],v = 6: [1, 1]+[2]t,v = 8: [1, 1]+[1, 2]+[1, 4]+[2, 2]+[2, 5]+[3, 3]+[4, 4],v = 10: [1, 2]+[4, 4]+[3, 3]13+[2]t,v = 12: [1, 2]+[2, 3]+[5, 6]+[4, 4]23 ,

Circular neighbor-balanced designs using cyclic shifts 2253

v = 14: [1, 3]6+[2, 5]5+{[2]+[5]+[6]}t,v = 16: [1, 3]+[1, 6]+[2, 3]+[2, 6]+[5]t,v = 18: [1, 2]+[1, 7]+[2, 3]+[4, 5]+[4, 6]+[6]t,v = 20: [1, 8]3+[1, 4]3+[2, 5]3+[2, 6]3+[3, 4]3+[3, 6]2+{[3]+[6]+[9]}t.

Series 4. For k = 4, v is prime and v−14 is not a multiple of 4, the sets of shifts to construct

CNBDs are given as: [1, 1, 1] + [2, 2, 2] + · · · + [v − 1, v − 1, . . . , v − 1].

Series 5. For k = 4, the set of shifts for the remaining CNBDs of Series 4 are given below:v = 5: [1, 3, 4],v = 6: [1, 1, 2]+[2, 2, 3]+[1, 3, 5]12 ,v = 8: [1, 1, 3]+[2, 4, 6]12+[2, 2, 2]14 ,v = 9: [2, 3, 5],v = 10: [1, 2, 4]+[1, 4, 2]+[1, 6, 5]+[1, 7, 4]+[3, 5, 7]12 ,v = 12: [2, 7, 5]+[5, 2, 3]+[1, 7]t,v = 13: [1, 2, 3]+[2, 3, 9]+[5, 5, 7],v = 14: [1, 2, 1]+[1, 2, 6]+[1, 4, 2]+[2, 3, 4]+[3, 3, 3]+[6, 6, 6]+[5, 7, 9]12 ,v = 15: [1, 2, 3]+[1, 2, 5]2+[2, 3, 6]+[3, 3, 5]+[4, 4, 8]+[6, 6, 8],v = 16: [1, 6, 5]+[3, 3, 7]+[4, 2, 10]+{[6]+[11]}t,v = 17: [2, 3, 4]+[5, 6, 7],v = 18: [2, 3, 4]4+[5, 6, 7]3+{[5, 6]+[1, 7]}t,v = 20: [1, 2, 5]+[2, 3, 4]+[4, 3, 6]+[6, 7, 8]+[9, 10, 11]12+[5, 5, 5]14 ,v = 21: [1, 4, 6]+[1, 4, 7]+[2, 3, 5]+[2, 5, 6]+[7, 8, 9],v = 22: [1, 4, 6]2+[1, 4, 7]2+[2, 3, 5]2+[2, 5, 6]2+[7, 8, 9]+{[3, 9]+[7, 8]}t,v = 24: [1, 2, 4]+[1, 3, 9]+[2, 3, 5]+[4, 5, 7]+[9, 10, 11]+[8, 12, 16]12+[6, 6, 6,]14 ,v = 25: [24, 24]+[3, 6, 7]+[10, 11, 12],v = 33: [1, 9, 5]+[2, 4, 11]+[3, 10, 6]+[8, 26, 12],v = 41: [3, 1, 15]+[4, 7, 14]+[5, 17, 21]+[6, 8, 18]+[10, 30, 29].

Series 6. For k = 5, v is prime and v−15 is not an integer, the sets of shifts to construct

CNBDs will be [1, 1, 1, 1] + [2, 2, 2, 2] + · · · + [v − 1, v − 1, v − 1, v − 1].

Series 7. For k = 5, the sets of shifts to construct remaining CNBDs of Series 6 are givenbelow:

v = 6: [1, 2, 2, 3]+[1, 2, 5, 3],v = 8: [1, 1, 1, 1]+[2, 2, 2, 3]3+[3, 3, 3, 3]+[3, 1, 1, 4]+[4, 2, 3, 4],v = 9: [1, 1, 2, 3]+[1, 3, 3, 4]+[2, 1, 3, 4]+[3, 4, 4, 2],v = 10: [1, 2, 3, 5]+[1, 3, 4]t,v = 11: [1, 9, 4, 3],v = 12: [1, 1, 1, 1]+[2, 2, 2, 2]+[3, 3, 3, 3]+[4, 4, 4, 4]2+[5, 5, 5, 5]2+{[1, 1, 1]+[2, 2, 2]+[3,3, 3]+[1, 1, 2]+[3, 3, 4]}t,v = 14: [1, 1, 1, 1]+[2, 2, 2, 2]+[3, 3, 3, 3]+[4, 4, 4, 4]2+[5, 5, 5, 5]2+[6, 6, 6, 6]2+{[1, 1,1]+[2, 2, 2]+[3, 3, 3]+[1, 1, 2]+[3, 4, 5]}t,v = 15: [1, 4, 5, 7]+[3, 3, 3, 3]15+[6, 6, 6, 6]15 ,v = 16: [1, 8, 3, 7]+[5, 5, 6, 7]+[1, 2, 13, 12],

2254 Iqbal I et al.

v = 18: [1, 1, 1, 1]+[2, 2, 2, 2]2+[3, 3, 3, 3]+[4, 4, 4, 4]+[5, 5, 5, 5]2+[6, 6, 6, 6]2+[7, 7, 7,7]2+[8, 8, 8, 8]2+{[1, 1, 1]+[3, 3, 3]+[4, 4, 4]+[1, 1, 3]+[4, 4, 5]}t,v = 20: [1, 2, 3, 3]+[1, 7, 9, 10]+[5, 5, 6, 6]+[4, 4, 4, 4]25+[8, 8, 8, 8]25 ,v = 21: [1, 3, 8, 19]+[4, 6, 7, 9],v = 22: [1, 1, 1, 1]2+[2, 2, 2, 2]2+[3, 3, 3, 3]2+[4, 4, 4, 4]2+[5, 5, 5, 5]2+[6, 6, 6, 6]+[7, 7, 8,8]4+[9, 9, 9, 9]+[10, 10, 10, 10]{[2, 3, 6]+[6, 6, 6]+[7, 7, 6]+[10, 10, 10]2}t,v = 25: [3, 1, 2, 6]+[4, 18, 8, 9]+[5, 5, 5, 5]15+[10, 10, 10, 10]15 ,v = 31: [2, 20, 18, 10]+[4, 16, 2, 8]+[12, 17, 6, 24],v = 35: [5, 12, 20, 13]+[3, 1, 5, 9]+[7, 7, 7, 7]15+[14, 14, 14, 14]15 .

Series 8. For k = 6, v is prime and v−16 is not an integer, the sets of shifts for the construction

of CNBDs are given as: [1, 1, 1, 1, 1] + · · · + [v − 1, v − 1, v − 1, v − 1, v − 1].

Note. The above designs are neighbor-balanced at distances up to k − 1.

Series 9. The sets of shifts for remaining CNBDs of series 8 when k = 6 are given below:v = 8: [1, 1, 2, 2, 4]+{[1, 1, 1, 1]+[2, 2, 2, 2]+[3, 3, 3, 3]}t,v = 9: [1, 2, 4, 3, 7]12+[1, 2, 2, 5]t,v = 12: [1, 2, 3, 4, 5]+[2, 3, 5, 10]t,v = 13: [6, 3, 5, 9, 2],v = 25: [2, 1, 11, 8, 7]+[6, 3, 9, 15, 5],v = 31: [1, 4, 2, 5, 3]+[6, 7, 9, 8, 18]+[12, 16, 27, 11, 23].

Series 10. When k = v−12 , the sets of shifts for CNBDs are given below:

v = 7, k = 3: [1, 2],v = 9, k = 4: [2, 3, 5],v = 11, k = 5: [1, 3, 4, 5],v = 13, k = 6: [1, 2, 3, 5, 6],v = 15, k = 7: [1, 3, 4, 6, 8, 13],v = 17, k = 8: [2, 4, 5, 3, 6, 7, 16],v = 19, k = 9: [3, 2, 15, 5, 6, 7, 8],v = 21, k = 10: [1, 3, 2, 5, 4, 8, 6, 9, 14],v = 23, k = 11: [1, 2, 3, 4, 5, 6, 7, 8, 9, 13],v = 25, k = 12: [1, 3, 4, 5, 6, 8, 7, 9, 10, 11, 23],v = 27, k = 13: [1, 3, 4, 2, 6, 8, 9, 7, 12, 11, 10, 13].

Series 11. For k = v and v is prime, the sets of shifts for the construction of CNBDs will be[1, 1, . . . , 1] 1v + [2, 2, . . . , 2] 1v + · · · + [v−1

2 , v−12 , . . . , v−1

2 ] 1v .

Note. The above designs are also neighbor-balanced at distances up to k − 1.

Series 12. When k = v and v is multiple of 4 and the preceding treatment is a primenumber, that is, v = 4, 8, 12, 20, 24, 32, 44, 48, 60, 72, 80, 84, . . ., then the following set of shiftsfor the construction of CNBDs are follows:

[1, 2, 3, . . . ,

v − 42

,v − 2

2,v − 2

2,v − 4

2, . . . , 3, 2, 1

].

Series 13. When k = v, the sets of shifts for remaining CNBDs of series 12 are given below:

Circular neighbor-balanced designs using cyclic shifts 2255

v = 4: [1, 1]t,

v = 6: [1, 3, 3, 1]t,

v = 9: {[1, 2, 3, 4, 5, 6]12}t,v = 10: [1, 2, 3, 5, 5, 3, 2, 1]t,

v = 14: [5, 4, 1, 2, 3, 4, 5, 6, 3, 1, 6, 2]t,

v = 15: {[2, 4, 6, 13, 11, 9, 7, 5, 3, 1, 8, 10, 12]12}t,v = 16: [6, 1, 1, 3, 3, 5, 5, 4, 4, 8, 6, 2, 2, 7]t,

v = 18: [1, 2, 3, 4, 5, 6, 7, 8, 7, 5, 2, 6, 3, 4, 1, 8]t.

Series 14. For k = v − 1, the sets of shifts for CNBDs are given below:

v = 4, k = 3: [1, 2],

v = 5, k = 4: [1, 3, 4],

v = 6, k = 5: [2, 2, 2, 2]15+[1,3,4]t,

v = 7, k = 6: [1, 1, 2, 2, 4],

v = 8, k = 7: [1, 2, 3, 4, 3, 2],

v = 9, k = 8: [2, 2, 4, 4, 3, 1, 3],

v = 10, k = 9: [1, 1, 1, 1, 1, 1, 1, 1]19+[2, 2, 4, 4, 3, 1, 3]t,

v = 11, k = 10: [1, 1, 3, 2, 2, 5, 3, 4, 5],

v = 12, k = 11: [1, 2, 3, 4, 6, 5, 5, 3, 2, 4],

v = 13, k = 12: [5, 9, 1, 2, 3, 2, 3, 4, 7, 1, 8],

v = 14, k = 13: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 113+[6, 1, 2, 2, 3, 3, 1, 4, 4, 6, 5, 5]t,

v = 15, k = 14: [3, 1, 4, 6, 7, 5, 5, 9, 2, 1, 4, 3, 2],

v = 16, k = 15: [5, 2, 4, 1, 2, 3, 3, 4, 5, 6, 7, 8, 7, 6],

v = 17, k = 16: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 117+[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

12, 13, 14]t,

v = 18, k = 17: [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8] 118+[8, 1, 2, 3, 4, 5, 6, 7, 8, 7, 5,

2, 6, 3, 4, 1]t.

General note. In shift [1,8]t, t means add (v − 1) treatments in the last row, and the shift[1,8]3 means, use set of shift [1,8] three times.

5 Remarks

If k = [v−12 ] or k = [v−1

4 ], then it is particularly very easy to construct CNBDs with λ1. Theseresults are stated in the following two conjectures:

Conjecture 1. If k = [v−12 ], then a CNBD can be constructed using a single set of shifts.

Conjecture 2. If k = [v−14 ], then a CNBD can be constructed using two sets of shifts.

If we compare our CNBDs with the designs available in the literature, for example Rees[3],Hawang[5], Dey and Chakravarty[7], Hawang and Lin[8], Chandak[11] and Street[12], we find thatmany of our designs are new, at the same time our construction method is simple and straightforward. If we consider example 2.1 (see [1, p. 822]), the CNBD(5, 4, 5, 4) can be constructedusing the following sets of shifts [1, 1, 1, 1]15 +[2, 2, 2, 2]15 +[3, 3, 3, 3]15 +[4, 4, 4, 4]15 , which is alsopresent in Series 11. It is also interesting that all our proposed CNBDs are binary and majorityof them are treatment-balanced too.

2256 Iqbal I et al.

In this paper, we have constructed CNBDs using method of cyclic shifts, which are circularand neighbor-balanced. Further, many of CNBDs are treatment-balanced and M -balanced.The properties of such CNBDs can easily be obtained from the sets of shifts and also from theoff-diagonal elements of concurrence matrix. Due to circulant property, all these CNBDs areA-optimal and efficient. Further, these suggested CNBDs are circular, balanced and binary,hence they are also universally optimal due to Druilhet[19], Bailey and Druilhet[17], Filipiak andMarkiewicz[18] and Ai et al.[1].

Acknowledgements The authors wish to thank the editor, and the two referees for theirvaluable comments which led to improvements in the first version of this paper.

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