Some important classes of neighbor balanced designs in linear blocks of small sizes

Amin Khalid; Farrukh Shehzad; Azeem Ali; Rashid Ahmed

doi:10.1016/j.jksus.2017.10.001

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Original article

30 (

); 311-315

doi:

10.1016/j.jksus.2017.10.001

Some important classes of neighbor balanced designs in linear blocks of small sizes

Amin Khalid^{^a}, Farrukh Shehzad^{^b}, Azeem Ali^{^c}, Rashid Ahmed^{^b,⁎}

a Allama Iqbal Open University Islamabad, Pakistan

b Department of Statistics, The Islamia University of Bahawalpur, Pakistan

c Department of Statistics, Govt. College University Lahore, Pakistan

⁎Corresponding author. rashid701@hotmail.com (Rashid Ahmed)

Received: 2016-10-24, Accepted: 2017-10-3, Published: 2017-07-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

Neighbor balanced designs are useful to balance out the neighbor effects in field of agriculture, serology, agro forestry, industry, etc. In most of the agriculture experiments blocks are formed in a line and therefore, neighbor balanced designs are required in linear blocks. In this article some classes of first order neighbor balanced designs are presented in linear blocks of size three and four. A method to construct the second order neighbor balanced designs through two minimal first order neighbor balanced designs in linear binary blocks of size three is also developed here.

Keywords

Neighbor effects

Linear blocks

First order neighbor balanced designs

Second order neighbor balanced designs

Minimal designs

05B05

62K10

1 Introduction

If v treatments are arranged in b linear blocks of size k such that each unordered pair of adjacent treatments appears an equal number of times, say λ₁′, designs are called first order neighbor balanced designs in linear blocks. If λ₁′ = 1 then such designs are called minimal first order neighbor balanced designs in linear blocks. Kiefer and Wynn (1981) introduced an algorithm to construct the neighbor balanced designs (NBDs) in complete linear blocks. Cheng (1983) generated NBDs in linear blocks for different cases. Azais et al. (1993) constructed NBDs in complete blocks, in k = v − 1 and partially neighbor balanced designs in linear blocks. Jacroux (1998) constructed NBDs for all v having blocks of size 3 which are efficient under standard intrablock analysis as well as when experimental units adjacent within blocks are correlated. Tomar et al. (2005) constructed neighbor balanced block designs using Mutually Orthogonal Latin Squares (MOLS) and compared their designs with complete block designs balanced for neighbor effects. Ahmed (2010) constructed NBDs in linear blocks for k even, k odd &two different block sizes k₁ and k₂. Ahmed and Akhtar (2011) constructed NBDs in linear blocks of equal sizes for (i) v = 4i + 1, i integer, k = 3 with λ′ = 1, (ii) v = 2i + 1, i (>1) odd, k = 3 with λ′ = 2, and (iii) v = 2i + 1 (prime) and k < v. They also constructed these designs in linear blocks of unequal sizes for (i) v = 4i–1; in k₁ = 3 and k₂ = 2, λ′ = 1, (ii) v = 4i + 2; in k₁ = 3 and k₂ = 2. Ahmed et al. (2013) developed some infinite series to generate minimal neighbor balanced designs for two and three different sizes in linear blocks. They also constructed generalized neighbor designs (GN₂-designs) in proper linear blocks. They developed following infinite series of minimal neighbor balanced designs in linear blocks. Shahid et al. (2017) constructed some important classes of generalized neighbor designs in linear blocks for four different cases. Minimal designs are always considered as the most economical. In this article, some infinite series are developed to generate the first order neighbor balanced designs for linear blocks of size 3 and 4. Catalogues are also presented of proposed designs. Two series are also developed to generate second order neighbor balanced designs through two minimal first order neighbor balanced designs in linear binary blocks of size three. These designs are constructed using method of cyclic shifts which is explained in Section 2. It is important to note that all proposed neighbor designs, in this article, are assumed balanced for non-directional neighbor effect. Non-directional neighbor effect means same effect from right or left neighbor and direction of neighbor does not matter.

2 Models and notations

Consider a class of designs $Ω_{(v, b, k)}$ with v treatments grouped in b circular blocks of k experimental units per block.

Definition 1

A NBD at distance 1 is a binary design in which each treatment appears as first order neighbor $λ^{'}$ times to all other treatments.

Definition 2

A NBD at distance 2 is a binary design in which each treatment appears as first order neighbor $λ_{1}^{'}$ times to all other treatments and each treatment appears as second order neighbor $λ_{2}^{'}$ . times to all other treatments.

NBD at distance 1 & 2 are constructed under the following models 1 & 2 respectively:

(1)

y_{ij} = μ + β_{i} + τ_{d (i, j)} + λ_{1 d (i, j \pm 1)} + ε_{ij} i = 1, 2, . \dots, b; j = 1, 2, \dots, k

(2)

y_{ij} = μ + β_{i} + τ_{d (i, j)} + λ_{1 d (i, j \pm 1)} + λ_{2 d (i, j \pm 1)} + ε_{ij} i = 1, 2, \dots, b; j = 1, 2, \dots, k

Models (1) and (2) can be written in vector as

(3)

Y_{d} = 1 μ + B β + T_{d} τ + U_{1 d} λ_{1} + ε

(4)

Y_{d} = 1 μ + B β + T_{d} τ + U_{1 d} λ_{1} + U_{2 d} λ_{2} + ε

where

Y

and

1

are vectors of observations and 1’s each of order (bk × 1) respectively,

T_{d}

U_{1 d}

and

U_{2 d}

are incidence matrices of order (bk × v) for treatment, first order neighbor and second order neighbor effects respectively, and

B

is the (bk × b) incidence matrix for block effects. The vectors

β, τ, λ_{1}, λ_{2}

are the parameters of respective effects.

ε

is the vector of error term with mean vector

0

and variance-covariance matrix

σ^{2} I

. The complete information matrix for model (3) is

C_{d} = [\begin{matrix} 1^{'} 1 & 1^{'} B & 1^{'} T_{d} & 1^{'} U_{1 d} \\ B^{'} 1 & B^{'} B & B^{'} T_{d} & B^{'} U_{1 d} \\ T_{d}^{'} 1 & T_{d}^{'} B & T_{d}^{'} T_{d} & T_{d}^{'} U_{1 d} \\ U_{1 d}^{'} 1 & U_{1 d}^{'} B & U_{1 d}^{'} T_{d} & U_{1 d}^{'} U_{1 d} \end{matrix}]

where

1^{'} 1 = bk = rv = n

, For an equireplicate designs with constant block size k;

B^{'} B = {kI}_{b}

and

T_{d}^{'} T_{d} = {rI}_{v}

T_{d}^{'} B = N

U_{1 d}^{'} B = 2 N

. Let

T_{d}^{'} U_{1 d} = L

and

U_{1 d}^{'} U_{1 d} = M

, the complete information matrix becomes

C_{d} = [\begin{matrix} n & 1^{'} B & 1^{'} T_{d} & 1^{'} U_{1 d} \\ B^{'} 1 & {kI}_{b} & N^{'} & 2 N^{'} \\ T_{d}^{'} 1 & N & {rI}_{v} & L \\ U_{1 d}^{'} 1 & 2 N & L & M \end{matrix}]

After imposing restrictions and simplification, the joint information matrix for treatment and neighbor effects is $C_{τ u} = [\begin{matrix} T_{d}^{'} T_{d} - {(T_{d}^{'} B_{d} K)}^{- 1} (B_{d}^{'} T_{d}) & T_{d}^{'} U_{1 d} - {(T_{d}^{'} B_{d} K)}^{- 1} (B_{d}^{'} U_{1 d}) \\ U_{1 d}^{'} T_{d} - (U_{1 d}^{'} B_{d}) K^{- 1} (B_{d}^{'} T_{d}) & U_{1 d}^{'} U_{1 d} - (U_{1 d}^{'} B_{d}) K^{- 1} (B_{d}^{'} U_{1 d}) \end{matrix}]$ $C_{τ u} = [\begin{matrix} {rI}_{v} - (1 / k) {NN}^{'} & L - (2 / k) {NN}^{'} \\ L - (2 / k) {NN}^{'} & M - (4 / k) {NN}^{'} \end{matrix}]$

Then information matrix for treatment is $C_{τ} = A - {BD}^{- 1} B^{'}$ where $A = {rI}_{v} - (1 / k) {NN}^{'}$ $B = L - (2 / k) {NN}^{'}$ $D = M - (4 / k) {NN}^{'}$

Here ${NN}^{'}$ is the treatment concurrence matrix whose diagonal elements are repetitions of each treatment and off-diagonal elements are the number of times two treatments appear together in same blocks. L is the incidence matrix of treatments versus neighbors (left and right). Diagonal elements of L matrix, for a design in which no treatment appears as neighbor to itself, are zero and off-diagonal matrix are the number of times a pair of treatments appear as neighbor to each other in same blocks. For further detail see Iqbal et al. (2006, 2009). To achieve a NBD, all off-diagonal elements of matrix L must be same. If off-diagonal elements of matrix L contain two or more distinct values, the design is known as generalized neighbor design. Similarly, if off-diagonal elements of concurrence matrix ${NN}^{'}$ are same then the design is BIBD otherwise PBIBD.

Information matrix for neighbor effect in model (3) and information matrices for model (4) can be derived accordingly.

According to Hinkelmann and Kempthorne (2005), average variance of treatment contrast is a function of information matrix as $\underset{i \neq i^{'}}{av .} var {({\hat{τ}}_{i} - {\hat{τ}}_{i^{'}})}_{IBD} = 2 σ_{e (IBD)}^{2} {(v - 1)}^{- 1} \sum_{i = 1}^{v - 1} d_{i}^{- 1}$ where d_i is the ith eigenvalue of $C_{τ}$ . Eigen values of $C_{τ}$ can be calculated using R-language.

3 Method of construction and efficiency factor

3.1

3.1 Method of cyclic shifts

Method of cyclic shifts introduced by Iqbal (1991) is simplified here to construct neighbor balanced designs only in linear blocks”. v treatments are labeled as 0, 1, 2, …, v − 1 under rule I and II below.

Rule I: Let S_j = [q_j1, q_j2, …, q_j(k-1)] be a set of shifts where 1 ≤ q_ji ≤ v − 1. A design is first order neighbor balanced designs in linear blocks if each element of S_j along with its complement contains all elements 1, 2, …, v − 1 equally often, say, λ₁ times. In Rule I, complement of q_i is v − q_i.

Rule II: Let S_j = [q_j1, q_j2, …, q_j(k-2)]t be a set of shifts where 1 ≤ q_ji ≤ v − 2. A design is first order neighbor balanced designs in linear blocks if each element of S_j along with its complement contains all elements 1, 2, …, v − 2 equally often, say, λ₁ times. In Rule II, complement of q_i is v − 1 − q_i.

3.2

3.2 Efficiency factor (E_f)

In literature, neighbor designs in incomplete block designs are compared with (i) complete block designs and (ii) complete block design balanced for neighbor effects. Hinkelmann and Kempthorne (2005) suggested former approach to calculate relative efficiency, Efficiency Factor E_f. The approach allows comparing an incomplete block design (IBD) with CRD or RCBD and two competing IBDs with each other. The residual variances ( $σ_{c}^{2}$ ) is assumed same for both digns to be compared. For equal blocks of size k the information matrix of a model without and interference effect can be defined as: $C = R - \frac{1}{k} {NN}^{'}$

where, R is a diagonal matrix of order (v × v) with diagonal elements (r-number of replications). The matrix C has one Eigen value (root) $d_{v} = 0$ with normalized associated Eigenvector $D_{v} = (1 / v) J$ and the other non-zero roots will be $d_{1}, d_{2}, \dots, d_{v - 1}$ with orthonormal associated Eigenvectors $D_{1}, D_{2}, \cdot \cdot, D_{v - 1}$ . For a BIBD, $d_{1} = d_{2} = \cdot \cdot = d_{v - 1}$ whereas in PBIBD, number of different values of $d_{1}, d_{2}, \cdot \cdot, d_{v - 1}$ relate with association scheme. Consider same residual variances $σ_{e}^{2}$ for CRD and IBD;

Then $E = \frac{\underset{i \neq i^{'}}{av .} var {({\hat{τ}}_{i} - {\hat{τ}}_{i^{'}})}_{CRD / RCBD}}{\underset{i \neq i^{'}}{av .} var {({\hat{τ}}_{i} - {\hat{τ}}_{i^{'}})}_{IBD}} = \frac{2 / r}{2 / c} = \frac{2 / r}{2 / ({rE}_{f})} where σ_{e (CRD)}^{2} = σ_{e (IBD)}^{2}$

The quantity $c = {rE}_{f}$ is harmonic mean of non-zero Eigen values of matrix C. While comparing two competing IBDs having same number of observations (n), each IBD will have associated with an efficiency factor E_f, the IBD having higher E_f is considered as better design.

3.3

3.3 Upper bound (UB) for efficiency factor

In mathematics, an upper bound of a subset S of some partially ordered set P is an element which is greater than or equal to every element of S. The upper bound given by Hinkelmann and Kempthorne (2005) for an incomplete block designs is $E_{f} = \frac{(k - 1) v}{(v - 1) k} S .$

A balanced incomplete block design always achieve its upper bound equal to efficiency factor while partially balanced incomplete block design attain smaller E_f value than the upper bound.

4 First order neighbor balanced designs in linear blocks of size three

In this section, some infinite series to generate the first order neighbor balanced designs are developed in linear blocks of size three.

Series 4.1. Minimal first order NBDs can be constructed for v = 4i, i integer, k = 3 and λ₁ = 1 in i(v − 1) linear blocks from the following i sets of shifts.

S_j = [p, p + 1]; j = 1, …, i − 1 and p = 2j − 1.
S_i = [(v − 2)/2]t

Example 4.1

Following is minimal first order NBD for v = 8 and k = 3 with λ₁ = 1 generated through the sets of shifts [1, 2] and [3]t mod 7.

B1	B2	B3	B4	B5	B6	B7	B8	B9	B10	B11	B12	B13	B14
0	1	2	3	4	5	6	0	1	2	3	4	5	6
1	2	3	4	5	6	0	3	4	5	6	0	1	2
3	4	5	6	0	1	2	7	7	7	7	7	7	7

Series 4.2. First order NBDs can be constructed for v = 2i, i integer greater than 1 and k = 3 with λ₁ = 2 through the following sets of shifts mod v − 1.

S_j = [j, j]; j = 1, 2,…, i − 2.
S_i_-1 = [(v − 2)/2]t, S_i = [v/2]t

Example 4.2

First order NBD for v = 10 and k = 3 with λ₁ = 2 can be generated through the sets of shifts [1, 1], [2, 2],[3, 3], [4]t and [5]t mod 9.

5 First order neighbor balanced designs in linear blocks of size four

In this section, some infinite series to generate the first order neighbor balanced designs are developed in linear blocks of size four.

Series 5.1. Minimal first order NBDs can be constructed for v = 6i + 1,iinteger and k = 4 with λ₁ = 1 through the following i sets of shifts.

S_j = [3j − 2, 3j − 1, 3j]; j = 1, …, i.

Example 5.1

Minimal first order NBD for v = 13 and k = 4 with λ₁ = 1 can be generated through the sets of shifts [1, 2, 3] and [4, 5, 6] mod 13.

Series 5.2. First order NBDs can be constructed for v = 3i + 1 and k = 4 with λ₁ = 2 through the following sets of shifts mod v.

S_j₊₁ = [(3j)/2 + 1, (3j)/2 + 2, v − ((3j)/2 + 3)]; j = 0, 2, …, i.
S_j₊₁ = [3(j − 1)/2 + 3, 3(j − 1)/2 + 2, v − (3(j − 1)/2 + 3)]; j = 1, 3, …, i − 1.

Example 5.2

First order NBD for v = 7 and k = 4 with λ₁ = 2 can be generated through the sets of shifts [1, 2, 6] and [3, 2, 4] mod 7.

Series 5.3. Minimal first order NBDs can be constructed for v = 6i + 3 and k = 4 with λ₁ = 1 through the following sets of shifts mod v.

S_j₊₁ = [3j + 1, v − (3j + 2), 3j + 3]; j = 0, 1, …, i − 1.
S_i₊₁ = [(v − 1)/2, (v − 1)/2, (v − 1)/2](1/3) (Every third block be taken from).

Example 5.3

Minimal first order NBD for v = 15 and, k = 4 with λ₁ = 1 can be generated through the sets of shifts [1, 13, 3] and [4, 10, 6] and [7, 7, 7](1/3) (Consider every third block) mod 15.

Series 5.4. Minimal first order NBDs can be constructed for v = 6i + 4 and k = 4 with λ₁ = 1 through the following sets of shifts mod v − 1.

S_j₊₁ = [3j + 1, v − (3j + 2), 3j + 3]; j = 0, 1, …, i − 1.
S_i₊₁ = [(v − 2)/2, v/2, (v + 2)/2](1/2)

Example 5.4

Minimal first order NBD for v = 16 and k = 4 with λ₁ = 1 can be generated through the sets of shifts [1, 14, 3], [4, 11, 6] and [7, 8, 9](1/2) mod 16.

Series 5.5. Minimal first order NBDs can be constructed for v = 3i, i even and k = 4 with λ₁ = 1 through the following sets of shifts mod v − 1.

S_j = [p, p + 1, p + 2]; j = 1, …, c − 1 and p = 3j − 2.
S_c = [(v − 4)/2, (v − 2)/2]t, where c = i/2

Example 5.5

Minimal first order NBD for v = 12 and k = 4 with λ₁ = 1 can be generated through the sets of shifts [1, 2, 3] and [4, 5]t mod 11.

Series 5.6. First order NBDs can be constructed for v = 6i + 2 and k = 4 with λ₁ = 3 through the following sets of shifts mod v − 1.

S_j = [j, j, j]; j = 1, …, 3i − 2.
S₃_i_-1 = [(v − 4)/2, (v − 4)/2]t,
S₃_i = [(v − 4)/2, (v − 2)/2]t,
S₃_i₊₁ = [(v − 2)/2, (v − 2)/2]t

Example 5.6

First order NBD for v = 8 and k = 4 with λ₁ = 3 can be generated through the sets of shifts [1, 1, 1] + [2, 2]t + [2, 3]t + [3, 3]t mod 7.

A catalogue of First order NBD is developed for linear blocks of size three using Series 3.1 & 3.2 and of size four using series 4.1, 4.2, …, 4.6 which is given as Supplementary Material.

6 Second order NBD in linear blocks of size three

In this Section, a method to construct the second order neighbor balanced designs through combining two minimal first order neighbor balanced designs in linear binary blocks of size three is developed.

A design is called second order neighbor balanced in linear blocks if each unordered pair of distinct treatments appears:

An equal number of times, say, λ₁ as first order/ adjacent neighbors, and
An equal number of times, say, λ₂ as second order neighbors.

A minimal second order neighbor balanced designs will be with λ₁ = 2 and λ₂ = 1.

Theorem 6.1

If v = 4i + 1, i odd then second order neighbor balanced designs with λ₂ = 1 can be constructed in linear binary blocks of size three by combing two first order neighbor balanced minimal designs through the following sets of shifts.

Design I (first order neighbor balanced minimal design):

S_j₊₁ = [2j + 1, 2j + 2]; j = 0, 1, …, i − 1.

Design II (first order neighbor balanced minimal design):

S_j₊₁ = [(2j + 2), 2j + 3]; j = 0, 1, …, i − 2 where j ≠ (i − 3)/2.
S_i_-₁ = [3i + 2, i] for i > 1
S_i = [1, 2i + 1]

Required Design (first-order and second order neighbor balanced minimal design):

S_i₊₁ = [2j + 1, 2j + 2]; j = 0, 1, …, i − 1.
S_j₊ _i₊₁ = [2j + 2, 2j + 3]; j = 0, 1, …, i − 2 where j ≠ (i − 3)/2.
S₂_i_-₁ = [3i + 2, i]
S₂_i = [1, 2i + 1] for i > 1

Example 6.1

For v = 5 the following are two different first order neighbor balanced designs in linear binary blocks of size three where each pair of distinct treatments appears twice as first order neighbors.

Design I					Design II
B₁	B₂	B₃	B₄	B₅	B₁	B₂	B₃	B₄	B₅
0	1	2	3	4	0	1	2	3	4
1	2	3	4	0	1	2	3	4	0
3	4	0	1	2	4	0	1	2	3

Combing these two designs, we get required design which is also second order neighbor balanced design with λ₁ = 2 and λ₂ = 1

B1	B2	B3	B4	B5	B6	B7	B8	B9	B10
0	1	2	3	4	0	1	2	3	4
1	2	3	4	0	1	2	3	4	0
3	4	0	1	2	4	0	1	2	3

Theorem 6.2

If v = 8i + 1, i be an integer then second order neighbor balanced designs λ₂ = 1 can be constructed in linear binary blocks of size three by combing two minimal first order neighbor balanced designs through the following sets of shifts.

Design I (first order neighbor balanced minimal design):

S_j₊₁ = [2j + 1, 2j + 2]; j = 0, 1, …, 2i − 1.

Design II (first order neighbor balanced minimal design):

S_j₊₁ = [2j + 2, 2j + 3]; j = 0, 1, …, 2i − 2 where j ≠ i − 1.
S_2i_-₁ = [6i + 1, 2i + 1] S2i = [1, 4i].

Required Design (first-order and second order neighbor balanced minimal design):

S_j₊₁ = [2j + 1, 2j + 2]; j = 0, 1, …, 2i − 1.
S_j₊_2t+1 = [2j + 2, 2j + 3]; j = 0, 1, …, 2i − 2 where j ≠ i − 1.
S₄_i_-₁ = [6i + 1, 2i + 1] S₄_i = [1, 4i].

Example 6.2

For v = 9 the following are two different first order neighbor balanced designs in linear binary blocks of size three where each pair of distinct treatments appears twice as first order neighbors.

Design I
B1	B2	B3	B4	B5	B6	B7	B8	B9	B10	B11	B12	B13	B14	B15	B16	B17	B18
0	1	2	3	4	5	6	7	8	0	1	2	3	4	5	6	7	8
1	2	3	4	5	6	7	8	0	3	4	5	6	7	8	0	1	2
3	4	5	6	7	8	0	1	2	7	8	0	1	2	3	4	5	6

Design II
B₁	B₂	B₃	B₄	B₅	B₆	B₇	B₈	B₉	B₁₀	B₁₁	B₁₂	B₁₃	B₁₄	B₁₅	B₁₆	B₁₇	B₁₈
0	1	2	3	4	5	6	7	8	0	1	2	3	4	5	6	7	8
1	2	3	4	5	6	7	8	0	3	4	5	6	7	8	0	1	2
3	4	5	6	7	8	0	1	2	7	8	0	1	2	3	4	5	6

Combing these two designs, we get required design which is also minimal second order neighbor balanced design with λ₁ = 2 and λ₂ = 1.

B1	B2	B3	B4	B5	B6	B7	B8	B9	B10	B11	B12
0	1	2	3	4	5	6	7	8	0	1	2
1	2	3	4	5	6	7	8	0	3	4	5
3	4	5	6	7	8	0	1	2	7	8	0

B13	B14	B15	B16	B17	B18	B19	B20	B21	B22	B23	B24
3	4	5	6	7	8	0	1	2	3	4	5
6	7	8	0	1	2	1	2	3	4	5	6
1	2	3	4	5	6	3	4	5	6	7	8

B25	B26	B27	B28	B29	B30	B31	B32	B33	B34	B35	B36
6	7	8	0	1	2	3	4	5	6	7	8
7	8	0	3	4	5	6	7	8	0	1	2
0	1	2	7	8	0	1	2	3	4	5	6

Acknowledgment

Authors are highly thankful to the Editor and Reviewers for their valuable suggestions and corrections.

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