Universally Optimal One-sided Circular Neighbour-balanced Designs with k = 5

Abstract

In many experiments, especially in agriculture and horticulture to some extent, the response from a plot in a block is affected by treatments on forward (backward) neighbour plots in the same block. Under such circumstances, one-sided circular neighbour-balanced design (one-sided CNBD) has wide applications. However, in concern with the universal optimality, there are very limited number of known series of one-sided CNBD’s. The purpose of this article is to present a new series of universally optimal one-sided CNBD with block size 5 where if a treatment appears repeatedly in a block, all of them are in a series of adjacent plots in the block.

Keywords

One-sided circular neighbour-balanced designs universally optimal design equivalence class of sequences optimal sequence

AMS 2000 subject classification: Primarily 62K05, 62K10, 62K99.

1. Introduction

Neighbour effects are studied in different areas of design of experiment under different models. Especially, in agriculture or allied areas, where the treatment applied to one experimental plot may influence the response of the only following neighbour plot as well as the plot to which it is applied. Bailey^[1] proposes a particular type of design that accounts for one-sided neighbour effects. In the experiment of cereal crops, sunflowers and others, where tall varieties may shade the plot on their north side and influence the response of the plot, such design is called for. Also, the application of such design lies in pesticide or fungicide experiments where some portion of the treatment applied may spread to the neighbour plot immediately down wind and spores from untreated plots may occur. The blocks of such designs are in linear ridges (one dimensional) [Welham et al.^[2]] where a treatment on a plot influences the response of the preceding (or following ) neighbour plot. The contributions of the literatures of Azais et al.,^[3] Smart et al.,^[4] Langton^[5] and David and Kempton^[6] have made a wider expansion in characterization and construction of neighbour effect designs. Bailey^[1] has developed such designs concerned with the study of one-sided neighbour effects. Later on, the work of Bailey^[1] has been extended by Bailey and Druilhet,^[7] taking into account the effect of the treatment on the preceding plot, in addition to the effects of the treatments on the plot and following plot apart from the effect of the block (if any).

One-sided circular neighbour-balanced design (one-sided CNBD) is an arrangement of v treatments in b linear blocks of size k (not necessarily distinct) such that (a) each treatment is replicated r times; (b) every pair of distinct treatments has concurrence μ; and (c) every treatment is followed by every other treatment λ times (assuming that in every block, the last plot is followed by the first plot). It is denoted by one-sided CNBD (v, b, r, k, s, m, n₁, n₂, μ, λ). By denoting d(i,j), the treatment assigned to plot j of block i and d(i,0), the treatment to the border plot of block i d(i,0)=d(i,k). However, the direct effect of d(i,0) on plot 0 of block i is not counted for analysis purposes. Let Y_ij be the response from plot j of block i where the observations Y_ijs have common variance and are independent. For total effects, the linear model equations are Y_ij = β_i + τ_d(i,j) + λ_d(i,j-1) + ξ_d(i,j) where β_i, τ_d(i,j), λ_d(i,j-1) and ξ_d(i,j) are the fixed effect of block i, the fixed effect of treatment d(i,j), the fixed left neighbour effect of treatment d(i,j − 1) and the random error of d(i,j) such that E(ξ_d(i,j)) = 0. The blocks of these designs are circular in theoretical construction of the design. However, they are linear by physical position in the experimental field, after having recommended a plot known as border plot before the first plot of each block, to which the treatment already applied to the last plot is applied. An example of one-sided CNBD (v = 3, b = 6, r = 6, k = 3, μ = 4, λ = 2) is given below, standing for by [α] the border plot receiving treatment α. ([a], c, a, a), ([c], a, c, c), ([a], b, a, a), ([b], a, b, b), ([b], c, b, b), ([c], b, c, c).

A design d with its information matrix, C_d, completely symmetric, is said to be universally optimal over a class Ω of competent designs, if trace(C_d) = max_dεΩ trace(C_d). While searching for the universally optimal among the competent designs in the class Ω, we look for optimal equivalence class of sequences in blocks of the design, instead of max_dεΩ trace(C_d), since equally often optimal occurrence of each sequence of equivalence class of sequences in the same number of blocks in the design d, say, ensures the maximization of c(ξ). Two sequences of treatments in two blocks are equivalent if one of the sequences can be obtained from the other one by relabelling the treatments. If we denote by ξ the equivalence class of the sequence l on the block u of the design d, the trace of C_du is given by c(ξ) = tr(C_du) = ( $k - \frac{2}{k} \sum_{i = 1}^{v} g_{i}^{2} + \sum_{i = 1}^{v} h_{i}$ )/2 (see Bailey and Druilhet⁷).

where g_i is the number of occurrences of treatment i the sequence l on the block u , and h_i is the number of times that treatment i is on the left-hand side of itself in the same sequence l. If a sequence in an equivalence class ξ^* of sequences maximizes c(ξ) among other sequences in other equivalence classes ξ^** of sequences where sequences are of same number of elements, then the equivalence class ξ^* is said to be optimal. It is obvious that two sequences of same number of elements belonging to two distinct equivalence classes of sequences may be, both, optimal. The Theorem 10, page no. 1658, Bailey and Druilhet^[7] states that one-sided CNBD, so-called by Meitei^[8], is universally optimal for estimation of the total effects among all possible designs with same size, if the design has sequences in an optimal equivalence class of sequences equally often.

2. Characterization

The five conditions of Proposition 9, page no. 1657, proposed by Bailey and Druilhet^[7], are not sufficient, but necessary conditions for existence of an optimal equivalence class of sequences as it can be seen that the value of c(ξ); ξ = (a, a, a, b, b), is 1.4 whereas that of c(ξ^*); ξ^* = (a, b, c, d, d) , is 1.6(≠1.4) though both ξ and ξ^* satisfy these five conditions. Among all the possible sequences of treatments in blocks of size 5, the optimal one is presented below.

Proposition 2.1. If the sequence in block of one-sided CNBD with k = 5 is either (a, a, b, b, c), (a, a, b, c, c), (a, b, b, c, c), (c, b, b, a, a), (c, c, b, a, a), or (c, c, b, b, a), the sequence has the optimal value 1.7.

Proof. Since one-sided CNBDs are circular, any one of the sequences, namely (a, a, b, b, c), (a, a, b, c, c), (a, b, b, c, c), (c, b, b, a, a), (c, c, b, a, a), or (c, c, b, b, a) can be obtained from the others by relabelling the treatments. Further, among all the possible sequences of treatments in blocks of size 5, namely ξ₁ = (a, b, c, d, e), ξ₂ = (a, b, c, d, d), ξ₃ = (a, b, b, c, c) and ξ₄ = (a, a, b, b, b), the sequence ξ₃ gives the maximum value of c(ξ), that is, the optimal value 1.7.

Remark 2.1. From Proposition 2.1, it is necessary and sufficient for one-sided CNBD with k = 5 to be universally optimal that (a) the sequence of treatments in any block is either ξ₁, ξ₂, ξ₃, or ξ₄ and (b) every sequence appears the same number of times.

As n₁ + n₂ = s and mn₁ + (m + 1)n₂ = k = sq (say); q a positive integer, a condition(which is verifiable) for impossible optimal sequence, is obtained as given below.

Proposition 2.2. If k is a multiple of s, then no optimal sequence exists, where n₁, n₂ > 0.

3. Construction

For the construction of one-sided CNBD, Lemma 2.1, by Meitei^[8], is called for in this section. The terms and the notations to be used in the lemma are as follows. Given a set S of size k, {i₁, i₂, … , i_k}, the forward and the backward differences arising from this set are defined as follows : F = (i₂ − i₁, i₃ − i₂, … , i_k − i_k-1, i₁ − i_k) and B = (i₁ − i₂, i₂ − i₃, … , i_k-1 − i_k, i_k − i₁), respectively. Clearly, B_k = -F_k.

Let v (=6t + 1, for some t) be a prime or prime power and x be the primitive element of the Galois field, GF(v). Consider the 6t sets $C_{i}^{(1)}$ , $C_{i}^{(2)}$ , $C_{i}^{(3)}$ $C_{i}^{* (1)}$ , $C_{i}^{* (2)}$ , and $C_{i}^{* (3)}$ ; i = 0, 1, …, t - 1, as defined below.

C_{i}^{(1)} = {x^{i}, x^{i}, x^{2 t + i}, x^{2 t + i}, x^{4 t + i}}

(3.1)

C_{i}^{(2)} = {x^{i}, x^{i}, x^{2 t + i}, x^{4 t + i}, x^{4 r + i}}

(3.2)

C_{i}^{(3)} = {x^{i}, x^{2 + i}, x^{2 + i}, x^{4 + i}, x^{4 t + i}}

(3.3)

C_{i}^{* (1)} = {x^{4 t + i}, x^{2 t + i}, x^{2 t + i}, x^{i}, x^{i}}

(3.4)

C_{i}^{* (2)} = {x^{4 + i}, x^{4 + i}, x^{2 + i}, x^{i}, x^{i}}

(3.5)

C_{i}^{* (3)} = {x^{4 + i}, x^{4 + i}, x^{2 + i}, x^{2 + i}, x^{i}}

(3.6)

By developing these 6t sets under the reduction modulo of v, we have a result as follows:

Theorem 3.1. Developing $C_{i}^{(1)}$ , $C_{i}^{(2)}$ , $C_{i}^{(3)}$ , $C_{i}^{* (1)}$ , $C_{i}^{* (2)}$ and $C_{i}^{* (3)}$ under mod(v); v (=6t + 1, for some t) a prime or prime power, yields a new series of universally optimal one-sided CNBD (v = 6t + 1, b = 6t(6t + 1), r = 30t, k = 5, s = 3, m = 1, n₁ = 1, n₂ = 2, μ = 16, λ = 3).

Proof. For a given GF(v); v = 6t + 1, consider the t initial blocks of the second series of the Steiner’s triple system with parameters v = 6t + 1, b = t(6t + 1), r = 3t, k = 3, λ = 1, that is, {xⁱ, x^2t+i, x^4t+i}. Finding all possible 6 differences of elements in every block of these t blocks, it is seen that all the 6t differences arisen from all the possible pairs of elements in these t blocks are non-zero elements of GF(v) and each non-zero element of GF(v) appears once among the totality of the differences of elements arisen from these t blocks. Since $C_{i}^{(1)}$ , $C_{i}^{(2)}$ and $C_{i}^{(3)}$ are inducted from the initial blocks of the Steiner’s triple system in $(_{2}^{3})$ , that is, 3 possible ways such that only 2 out of 3 elements of the Steiner’s triple system appears twice and the remaining once and $C_{i}^{* (1)}$ , $C_{i}^{* (2)}$ and $C_{i}^{* (3)}$ are such the reverse of $C_{i}^{(1)}$ , $C_{i}^{(2)}$ and $C_{i}^{(3)}$ respectively, all the non-zero element differences of elements arisen from the 6t sets given in (3.1)–(3.6) are all the non-zero elements of the GF(v) each repeating 8 + 8, that is, 16 times.

In Steiner’s triple system, any two elements are one forward neighbour of another and also one backward neighbour of another. Thus, among the totality of the non-zero forward differences arisen from $C_{i}^{(1)}$ , $C_{i}^{(2)}$ , $C_{i}^{(3)}$ , $C_{i}^{* (1)}$ , $C_{i}^{* (2)}$ and $C_{i}^{* (3)}$ , every non-zero element of GF(v) occurs thrice. Further, in each of the six initial sets, there exist three distinct elements, two appearing twice and another once. By Lemma 2.1, proposed by Meitei^[8], the proof of the one-sided CNBD is completed.

All the sequences in the t sets $C_{i}^{(1)}$ given in Equation (3.1) are equal, and the sequence in $C_{i}^{(1)}$ for a particular i occurs in (6t + 1) blocks (developed from $C_{i}^{(1)}$ ) of the design. Thus, the sequence in occurs in t(6t + 1) blocks of the design. Similarly, each of the sequences in $C_{i}^{(2)}$ , $C_{i}^{(3)}$ , $C_{i}^{* (1)}$ , $C_{i}^{* (2)}$ and $C_{i}^{* (3)}$ occurs in t(6t + 1) blocks of the design. Further, from Proposition 2.1, we know that sequences in $C_{i}^{(1)}$ , $C_{i}^{(2)}$ , $C_{i}^{(3)}$ , $C_{i}^{* (1)}$ , $C_{i}^{* (2)}$ and $C_{i}^{* (3)}$ are equally optimal with c(ξ)’s optimal value 1.7 (each occurring in t(6t + 1) blocks). Hence, the design is universally optimal.

Remark 3.1. Since every sequence of treatments in the sets $C_{i}^{(1)}$ , $C_{i}^{(2)}$ , $C_{i}^{(3)}$ , $C_{i}^{* (1)}$ , $C_{i}^{* (2)}$ and $C_{i}^{* (3)}$ is often equally optimal in t(6t + 1) out of 6t(6t + 1) blocks of the one-sided CNBD given in Theorem 3.1, with the optimal value 1.7 and they are equivalent, recalling Theorem 10, proposed by Bailley and Druilhet^[7], the trace of the information matrix of the present one-sided CNBD is equal to b • optimal value of c(ξ), that is, 10.2t(6t + 1). Thus, increase in t can ensure to achieve a desired quantity of trace of information matrix from the present design.

An example of the design proposed in Theorem 3.1 is presented below.

Example 3.1. If t = 1, then v = 7 and the primitive element of the GF(7) is 3. Further,

$C_{0}^{(1)}$ = {3⁰, 3⁰, 3², 3², 3⁴} = {1, 1, 2, 2, 4},

$C_{0}^{(2)}$ = {3⁰, 3⁰, 3², 3⁴, 3⁴} = {1, 1, 2, 4, 4},

$C_{0}^{(3)}$ = {3⁰, 3², 3², 3⁴, 3⁴} = {1, 2, 2, 4, 4},

$C_{0}^{* (1)}$ = {3⁴, 3², 3², 3⁰, 3⁰} = {4, 2, 2, 1, 1},

$C_{0}^{* (2)}$ = {3⁴, 3⁴, 3², 3⁰, 3⁰} = {4, 4, 2, 1, 1} and

$C_{0}^{* (3)}$ = {3⁴, 3⁴, 3², 3², 3⁰} = {4, 4, 2, 2, 1}.

These 6 sets under the reduction module of 7, yield the blocks of a Universally Optimal One-sided CNBD (7, 42, 30, 5, 1, 1, 2, 16, 3) with the optimal value 1.7 of c(ξ).

Footnotes

Declaration of Conflicting Interests

Funding

The author received no financial support for the research, authorship and/or publication of this article.

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