Minimal neighbor designs in circular blocks of unequal sizes

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Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 141 (2011) 3681–3685

0378-37

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jspi

Minimal neighbor designs in circular blocks of unequal sizes

Farrukh Shehzad a,n, Mohammad Zafar Yab b, Rashid Ahmed c

a National College of Business Administration & Economics, Lahore, Pakistanb Department of Statistics, University of Gujrat, Pakistanc Department of Statistics The Islamia University of Bahawalpur, Pakistan

a r t i c l e i n f o

Article history:

Received 24 November 2010

Received in revised form

27 May 2011

Accepted 2 June 2011Available online 24 June 2011

Keywords:

Circular blocks

Neighbor designs

Minimal neighbor designs

Neighbor effects

58/$ - see front matter & 2011 Elsevier B.V. A

016/j.jspi.2011.06.002

esponding author. Tel.: þ92 321 6840205.

ail addresses: [email protected] (F. Sh

a b s t r a c t

Neighbor designs have their own importance in the experiments to remove the

neighbor effects where the performance of a treatment is affected by the treatments

applied to its adjacent plots. If each pair of distinct treatments appears exactly once as

neighbors, neighbor designs are called minimal. Most of the neighbor designs require a

large number of blocks of equal sizes. In this situation minimal neighbor designs in

unequal block sizes are preferred to reduce the experimental material. In this article

some series are presented to construct minimal neighbor designs in circular blocks of

unequal sizes.

& 2011 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3681

2. Construction of minimal neighbor designs for k¼4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3682






8. Construction of minimal neighbor designs for k¼10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3683



Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3685

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3685

1. Introduction

A block is called circular if its first and last units are considered as neighbors. Neighbor design (ND) is a group of circularblocks in which each pair of distinct treatments appears equally often, say l0 times. Rees (1967) introduced circularneighbor designs in serology and constructed minimal neighbor designs for k¼v and some cases for vr41 with kr10,where v is odd. Then Lawless (1971), Hwang (1973), Azais et al. (1993), Ai et al. (2007), Ahmed and Akhtar (2008), Akhtar

ll rights reserved.

ehzad), [email protected] (M. Zafar Yab), [email protected] (R. Ahmed).

www.elsevier.com/locate/jspi

dx.doi.org/10.1016/j.jspi.2011.06.002

mailto:[email protected]



dx.doi.org/10.1016/j.jspi.2011.06.002

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3681–36853682

and Ahmed (2009), Iqbal et al. (2009) and Ahmed et al. (2010) constructed circular neighbor balanced designs for severalconfigurations. Neighbor balanced designs satisfy fairly restrictive combinatorial constraints and therefore, most of suchdesigns require large number of blocks. To reduce the number of blocks, following recommendations have already beenconsidered in the literature.

(i)
By relaxing the condition of constancy of l0:Wilkinson et al. (1983) introduced partially neighbor balanced designs. Misra et al. (1991), Chaure and Misra (1996), Nutan(2007), Kedia and Misra (2008), Ahmed et al. (2009) and ZafarYab et al. (2010) constructed generalized neighbor designs.
(ii)
Using some extra treatment(s):Ahmed and Akhtar (2010) considered the reduction in number of blocks using some extra treatment(s), whoseresponse will not be included in the analysis.
In this article, the reduction in number of blocks is proposed using blocks of unequal sizes. For example, 855experimental units are required to construct neighbor balance design for v¼19 in circular blocks of size 5 while it can bedone with just 171 experimental units using k1¼5 and k2¼4, which saves 80% experimental material. This article dealswith the construction up to k¼12.

2. Construction of minimal neighbor designs for k¼4

Series 2.1. Minimal circular blocks ND can be constructed for v¼8tþ7 in t initial blocks of k1¼4 and one initial block ofk2¼3 by developing the following (tþ1) initial blocks cyclically mod v, where t is positive integer.

Ijþ1 ¼ ð0,v�ð4jþ2Þ,1,ð4jþ5ÞÞ; j¼ 0,1,. . .,t�1:

Itþ1 ¼ ð0,ðvþ3Þ=2,1Þ:


Series 3.1. Minimal circular blocks ND can be constructed for v¼8tþ3 in one initial block of k1¼5 and (t�1) initialblocks of k2¼4 by developing the following t initial blocks cyclically mod v, where t (41) is integer.

I1 ¼ ð0,1,v�1,2,6Þ,

Ijþ1 ¼ ð0,v� 4jþ1Þ,2,ð4jþ6Þð Þ; j¼ 1,2,. . .,t�1:


Series 4.1. Minimal circular blocks ND can be constructed for v¼12tþ8iþ1 in t initial blocks of k1¼6 and i initialblocks of k2¼4 by developing the following (tþ i) initial blocks cyclically mod v, where i and t are positive integers.

Ijþ1 ¼ ð0,v�ð6jþ1Þ,1,v�ð6jþ2Þ,2,ð6jþ7ÞÞ; j¼ 0,2,. . .,t�1:

Ijþ1 ¼ ð0,v�6j,2,ð6jþ5Þ,1,ð6jþ6ÞÞ; j¼ 1,3,. . .,t�1:

�
for t odd: Itþuþ1¼(0, v�(4uþ6t), 2, 4uþ6tþ5): u¼0, 1,y, i�1. � for t even: Itþuþ 1¼(0, v�(4uþ6tþ1), 1, 4uþ6tþ4); u¼0, 1,y, i�1.

Series 5.1. Minimal circular blocks ND can be constructed for v¼12tþ8iþ15 in one initial block of k1¼7, t initial blocksof k2¼6 and i initial blocks of k3¼4 by developing the following (tþ iþ1) initial blocks cyclically mod v, where i and t arepositive integers.

I1 ¼ ð0,v�1,1,4,8,2,7Þ,

Ijþ1 ¼ ð0,v�ð6jþ2Þ,1,v�ð6jþ3Þ,2,ð6jþ8ÞÞ; j¼ 1,3,. . .,t:

Ijþ1 ¼ ð0,v�ð6jþ1Þ,2,ð6jþ6Þ,1,ð6jþ7ÞÞ; j¼ 2,4,. . .,t:

�
for t odd: Itþuþ2¼(0, v�(4uþ6tþ7), 2, 4uþ6tþ12); u¼0, 1, y, i�1. � for t even: Itþuþ2¼(0, v�(4uþ6tþ8), 1, 4uþ6tþ11); u¼0, 1, y, i�1. � for t¼0: Ij¼(0, v�4j, 1, 4jþ3); j¼2, 3, y, (iþ1).
Corollary 5.1. As t and i relate to k2 and k3, respectively. When any one of these is set to be zero, it becomes the minimalND for remaining respective block sizes.

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3681–3685 3683


Series 6.1. Minimal circular blocks ND can be constructed for v¼16tþ12iþ8sþ1 in t initial blocks of k1¼8, i initial
blocks of k2¼6 and s initial blocks of k3¼4 by developing the following (tþ iþs) initial blocks cyclically mod v, where t, i
and s are positive integers.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v�ð8jþ4Þ,3,ð8jþ8ÞÞ; j¼ 0,1,. . .,t�1:

Itþuþ1 ¼ ð0,v�ð6uþ8tþ1Þ,1,v�ð6uþ8tþ2Þ,2,6uþ8tþ7Þ; u¼ 0,2,. . .,i�1:

Itþuþ1 ¼ ð0,v�ð6uþ8tÞ,2,6uþ8tþ5,1,6uþ8tþ6Þ; u¼ 1,3,. . .,i�1:

�
for i odd: Itþ iþwþ1¼(0, v�(4wþ8t þ6i), 2, 4wþ8tþ6iþ5); w¼0, 1, y, s�1. � for i even: Itþ iþwþ1¼(0, v�(4wþ8tþ6iþ1), 1, 4wþ8tþ6iþ4); w¼0, 1, y, s�1.
Corollary 6.1. As i and s relate to k2 and k3, respectively. When any one of these is set to be zero, it becomes the ND ofremaining respective block sizes.


Series 7.1. Minimal circular blocks ND can be constructed for v¼16tþ12iþ8sþ19 in one initial block of k1¼9, t initialblocks of k2¼8, i initial blocks of k3¼6 and s initial blocks of k4¼4 by developing the following (tþ iþsþ1) initial blockscyclically mod v, where t, i and s are positive integers.

I1 ¼ ð0,v�1,1,4,8,3,9,2,10Þ,

Ijþ1 ¼ ð0,v�ð8jþ1Þ,2,v�ð8jþ2Þ,3,v�ð8jþ4Þ,4,8jþ10Þ; j¼ 1,2,. . .,t:

Itþuþ2 ¼ ð0,v�ð6uþ8tþ9Þ,2,6uþ8tþ14,1,6uþ8tþ15Þ; u¼ 0,2,. . .,i�1:

Itþuþ2 ¼ ð0,v�ð6uþ8tþ10Þ,1,v�ð6uþ8tþ11Þ,2,6uþ8tþ16Þ; u¼ 1,3,. . .,i�1:

�
for i odd: Itþ iþwþ2¼(0, v�(4wþ6iþ8tþ10), 1, 4wþ6uþ8tþ13); w¼0, 1, y, s�1. � for i even: Itþ iþwþ2¼(0, v�(4wþ6iþ8tþ9), 2, 4wþ6uþ8tþ14); w¼0, 1, y, s�1.
Corollary 7.1. As t, i and s relate to k2, k3 and k4, respectively. When any one, two or three of these set to be zero, itbecomes the ND of remaining respective block sizes.


Series 8.1. Minimal circular blocks ND can be constructed for v¼20tþ16iþ12sþ8pþ1 in t initial blocks of k1¼10, iinitial blocks of k2¼8, s initial blocks of k3¼6 and p initial blocks of k4¼4 by developing the following (tþ iþsþp) initialblocks cyclically mod v, where t, i, s and p are positive integers.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,. . .,t�1:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,. . .,t�1:

�
for t odd:
Itþuþ1 ¼ ð0,v�ð8uþ10tÞ,2,v�ð8uþ10tþ1Þ,3,v�ð8uþ10tþ3Þ,4,8uþ10tþ9Þ; u¼ 0,1,. . .,i�1:

Itþ iþwþ1 ¼ ð0,v�ð6wþ10tþ8iÞ,2,6wþ10tþ8iþ5,1,6wþ10tþ8iþ6Þ; w¼ 0,2,. . .,s�1:

Itþ iþwþ1 ¼ ð0,v�ð6wþ10tþ8iþ1Þ,1,v�ð6wþ10tþ8iþ2Þ,2,6wþ10tþ8iþ7Þ; w¼ 1,3,. . .,s�1:

Itþ iþ sþxþ1 ¼ ð0,v�ð4xþ10tþ8iþ6sÞ,2,4xþ10tþ8iþ6sþ5Þ; x¼ 0,1,. . .,p�1 for s even:Itþ iþ sþxþ1 ¼ ð0,v�ð4xþ10tþ8iþ6sþ1Þ,1,4xþ10tþ8iþ6sþ4Þ; x¼ 0,1,. . .,p�1 for s odd:

�
for t even:
Itþuþ1 ¼ ð0,v�ð8uþ10tþ1Þ,1,v�ð8uþ10tþ2Þ,2,v�ð8uþ10tÞþ4Þ,3,8uþ10tþ8Þ; u¼ 0,1,. . .,i�1:

Itþ iþwþ1 ¼ ð0,v�ð6wþ10tþ8iþ1Þ,1,v�ð6wþ10tþ8iþ2Þ,2,6wþ10tþ8iþ7Þ; w¼ 0,2,. . .,s�1:

Itþ iþwþ1 ¼ ð0,v�ð6wþ10tþ8iÞ,2,6wþ10tþ8iþ5,1,6wþ10tþ8iþ6Þ; w¼ 1,3,. . .,s�1:

Itþ iþ sþxþ1 ¼ ð0,v�ð4xþ10tþ8iþ6sÞ,2,4xþ10tþ8iþ6sþ5Þ; x¼ 0,1,. . .,p�1 for s odd:

Itþ iþ sþxþ1 ¼ ð0,v�ð4xþ10tþ8iþ6sþ1Þ,1,4xþ10tþ8iþ6sþ4Þ; x¼ 0,1,. . .,p�1 for s even:

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3681–36853684

Corollary 8.1. As i, s and p relate to k2, k3 and k4, respectively. When any one, two or three of these are set to be zero, itbecomes the ND of remaining respective block size(s).


Series 9.1. Minimal circular blocks ND for v¼20tþ16iþ12sþ8pþ23 can be constructed in one initial block of k1¼11, t

initial blocks of k2¼10, i initial blocks of k3¼8, s initial blocks of k4¼6 and p initial blocks of k5¼4 by developing thefollowing (tþ iþsþpþ1) initial blocks cyclically mod v, where t, i, s and p are positive integers.

I1 ¼ ð0,v�1,1,v�2,2,7,13,20,12,21,11Þ,

Ijþ1 ¼ ð0,v�ð10jþ2Þ,1,v�ð10jþ3Þ,2,v�ð10jþ4Þ,3,v�ð10jþ6Þ,4,ð10jþ12ÞÞ; j¼ 1,3,. . .,t:

Ijþ1 ¼ ð0,v�ð10jþ1Þ,2,v�ð10jþ2Þ,3,v�ð10jþ4Þ,4,v�ð10jþ5Þ,5,ð10jþ11ÞÞ; j¼ 2,4,. . .,t:

�
for t odd:
Itþuþ2 ¼ ð0,v�ð8uþ10tþ11Þ,2,v�ð8uþ10tþ12Þ,3,v�ð8uþ10tþ14Þ,4,8uþ10tþ20Þ; u¼ 0,1,. . .,i�1:

Itþ iþwþ2 ¼ ð0,v�ð6wþ10tþ8iþ11Þ,2,6wþ10tþ8iþ16,1,6wþ10tþ8iþ17Þ; w¼ 0,2,. . .,s�1

Itþ iþwþ2 ¼ ð0,v�ð6wþ10tþ8iþ12Þ,1,v�ð6wþ10tþ8iþ13Þ,2,6wþ10tþ8iþ18Þ; w¼ 1,3,. . .,s�1

Itþ iþ sþ xþ2 ¼ ð0,v�ð4xþ10tþ8iþ6sþ11Þ,2,4xþ10tþ8iþ6sþ16Þ; x¼ 0,1,. . .,p�1 for s even:

Itþ iþ sþ xþ2 ¼ ð0,v�ð4xþ10tþ8iþ6sþ12Þ,1,4xþ10tþ8iþ6sþ15Þ; x¼ 0,1,. . .,p�1 for s odd:

�
for t even:
Itþuþ2 ¼ 0,v�ð8uþ10tþ12Þ,1,v�ð8uþ10tþ13Þ,2,v�ð8uþ10tþ15Þ,3,8uþ10tþ19ð Þ; u¼ 0,1,. . .,i�1:

Itþ iþwþ2 ¼ ð0,v�ð6wþ8iþ10tþ12Þ,1,v�ð6wþ8iþ10tþ13Þ,2,6wþ8iþ10tþ18Þ; w¼ 0,2,. . .,i�1

Itþ iþwþ2 ¼ ð0,v�ð6wþ10tþ8iþ11Þ,2,6wþ10tþ8iþ16,1,6wþ10tþ8iþ17Þ; w¼ 1,3,. . .,i�1

Itþ iþ sþ xþ2 ¼ ð0,v�ð4xþ10tþ8iþ6sþ11Þ,2,4xþ10tþ8iþ6sþ16Þ; x¼ 0,1, . . .,p�1 for s odd:

Itþ iþ sþ xþ2 ¼ ð0,v�ð4xþ10tþ8iþ6sþ12Þ,1,4xþ10tþ8iþ6sþ15Þ; x¼ 0,1,. . .,p�1 for s even:

Corollary 9.1. As t, i, s and p relate to k2, k3, k4 and k5, respectively. When any one, two or more of these are set to be zero,it becomes the ND of remaining respective block sizes.


Series 10.1. Minimal circular blocks ND for v¼24tþ20iþ16sþ12pþ8qþ1 can be constructed in t initial blocks ofk1¼12, i initial blocks of k2¼10, s initial blocks of k3¼8, p initial blocks of k4¼6 and q initial blocks of k5¼4 by developingthe following (tþ iþsþpþq) initial blocks cyclically mod v, where t, i, s, p and q are positive integers.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ; j¼ 0,1,. . .,t�1:

Itþuþ1 ¼ ð0,v�ð10uþ12tþ1Þ,1,v�ð10uþ12tþ2Þ,2,v�ð10uþ12tþ3Þ,3,v�ð10uþ12tþ5Þ,4,0uþ12tþ11Þ; u¼ 0,2,. . .,i�1:

Itþuþ1 ¼ ð0,v�ð10uþ12tÞ,2,v�ð10uþ12tþ1Þ,3,v�ð10uþ12tþ3Þ,4,v�ð10uþ12tþ4Þ,5,0uþ12tþ10Þ; u¼ 1,3,. . .,i�1:

�
for i odd:
Itþ iþwþ1 ¼ ð0,v�ð8wþ12tþ10iÞ,2,v�ð8wþ12tþ10iþ1Þ,3,v�ð8wþ12tþ10iþ3Þ,4,8wþ12tþ10iþ9Þ; w¼ 0,1,. . .,s�1:

Itþ iþ sþxþ1 ¼ ð0,v�ð6xþ12tþ10iþ8sÞ,2,6xþ12tþ10iþ8sþ5,1,6xþ12tþ10iþ8sþ6Þx¼ 0,2,. . .,p�1:

Itþ iþ sþxþ1 ¼ ð0,v�ð6xþ12tþ10iþ8sþ1Þ,1,v�ð6xþ12tþ10iþ8sþ2Þ,2,6xþ12tþ10iþ8sþ7Þ; x¼ 1,3,. . .,p�1:

Itþ iþ sþpþyþ1 ¼ ð0,v�ð4yþ12tþ10iþ8sþ6pÞ,2,4yþ12tþ10iþ8sþ6pþ5Þ; y¼ 0,1,. . .,q�1 for p even:

Itþ iþ sþpþyþ1 ¼ ð0,v�ð4yþ12tþ10iþ8sþ6pþ1Þ,1,4yþ12tþ10iþ8sþ6pþ4Þ; y¼ 0,1,. . .,q�1 for p odd:

�
for i even:
Itþ iþwþ1 ¼ ð0,v�ð8wþ12tþ10iþ1Þ,1,v�ð8wþ12tþ10iþ2,2,v�ð8wþ12tþ10iþ4Þ,3,8wþ12tþ10iþ8Þ w¼ 0,1,2,3,. . .,s�1:

Itþ iþ sþxþ1 ¼ ð0,v�ð6xþ12tþ10iþ8sþ1Þ,1,v�ð6xþ12tþ10iþ8sþ2Þ,2,6xþ12tþ10iþ8sþ7Þ x¼ 0,2,4,. . .,p�1:

Itþ iþ sþxþ1 ¼ ð0,v�ð6xþ12tþ10iþ8sÞ,2,6xþ12tþ10iþ8sþ5,1,6xþ12tþ10iþ8sþ6Þ; x¼ 1,3,5,. . .,p�1:

Itþ iþ sþpþyþ1 ¼ ð0,v�ð4yþ12tþ10iþ8sþ6pÞ,2,4yþ12tþ10iþ8sþ6pþ5Þ; y¼ 0,1,. . .,q�1 for p odd:

Itþ iþ sþpþyþ1 ¼ 0,v� 4yþ12tþ10iþ8sþ6pþ1ð Þ,1,4yþ12tþ10iþ8sþ6pþ4ð Þ; y¼ 0,1,. . .,q�1 for p even:

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3681–3685 3685

Corollary 10.1. As i, s, p and q relate to k2, k3, k4 and k5 respectively. When any one, two or more of these are set to be zero,it becomes the ND of remaining respective block sizes.

Acknowledgment

Authors are thankful to the reviewer for his valuable comments, which contributed to the improvement of this article.

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Minimal neighbor designs in circular blocks of unequal sizes

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