1. Introduction

Some experimental material, may exhibit more than one source of variation that can be controlled by ordinary blocking and there may also be other types of relationships among several types of variation. A relationship that is often encountered in practice and which is of concern in this paper is nesting between two or more set of blocks. Example1 and 2 that utilised nesting relation are provided below as real life Animal and Plant experiments.

Example 1 (Plant Experiment) Preece [17], illustrated nesting, such that the half-leaves of a plant form the experimental units, on which a number of treatments were to be tested, where the number of treatments is more than the number of available half-leaves per plant. Clearly, one source of variation is due to variability among the plants. Further, leaves within a plant might exhibit variation due to their location on the upper, middle or lower branch of the same plant. Therefore, leaves within plants are nested nuisance factor, the nesting being within the plants. The half-leaves being the experimental units, there are two systems of blocks, leaves (which may be called sub-blocks) being nested within plants.

Example 2 (Animal Experiment) Generally littermates, or (animals born in the same day) are experimental units within a block. However, animals within the same litter may be varying in their initial body weight. If body weight is taken as another blocking factor, we have a system of nested blocks within a block. More examples and discussion of nesting, can be found in the literature, see for example Srivastana [20] and Morgan [13]

Definition 1 A NBIBD with parameters (v, r, b₁, b₂, k₁, k₂, λ₁, λ ₂, m) is an incomplete block design with v treatments, each replicated r times, within two systems of blocks such that:

i. the second system is nested within the first, with each block from the first system (called blocks) containing exactly m blocks of the second system (called sub-blocks);

ii. ignoring the sub-blocks leaves a BIB design with parameters v, b₁, r, k₁, λ ₁; and

iii. ignoring the blocks leaves a BIB design with parameters v, b₂, r, k₂, λ ₂

Theorem 1 Jimbo and Kuriki [11]. Consider a BIBD with parameters v*, b*, k* = v and also a BIBD design with (v, b₁, b₂, k), if the NBIBD is written using the treatments of each block of the BIB design, then the resulting design is a NBIB (v*, b₁b*, b₂, k).

A BIBD is resolvable (Abel and Furino [1]) and Bailey [2], if its set of blocks can be partitioned into subsets such that each subset is a replicate or resolution class, such that each subset contains each treatment exactly once.

Remark, a resolvable incomplete block design is a special case of nested incomplete block designs, with the main blocks being complete. In view of this fact, if a resolvable BIB design is taken as a particular case of NBIB design in Theorem 2, and the NBIB design wherein is the same as ones obtained by Dey et. al. [7]. NBIB designs has been studied by a number of other authors, including Preece [17], Jimbo and Kuriki [11], Bailey [3], Iqbal [9], Jimbo [10], Jimbo and Kuriki [11] and Dey et. al. [5]

Theorem 2 The existence of a NBIB design (v, b₁, b₂, k) and of a resolvable BIB design with parameters v* = b₂, b*, k* implies the existence of a NBIB (v, b₁b*, k*, k).

2. Method of Construction

In recent time, the development of the theory of designs has continued to exploit the advantages of advanced topics in algebra and combinatorics. Wen-Fong and Gunter [22] has shown how some specific classes of algebraic structures (planar near rings) give rise to efficient balanced incomplete block designs. Also, Morgan et al. [14], reviewed and extended mathematical aspect of nested balanced incomplete block design (NBIBD’s); isomorphism and automorphisms were defined for NBIBDs, and methods of construction were outlined. Peter, et al. [16], showed the necessary divisibility conditions for the existence of a 𝜎-resolvable BIBD (v, k, ʎ) as sufficient for large v. Saka et al. [19], developed a new method of construction of nested balanced incomplete block designs in which the resulting design schemes were of the type that harmonizes both the Series-I and Series-II of Rajender et al. [18]. Keerti and Vineeta [12], introduced a new method of construction of a series of Nested Balanced Incomplete Block Designs (NBIBDs) in which the inner blocks are constructed using Latin square. In this paper we present a new method of constructing NBIB designs called Coset-k² of NBIB designs of resolvable type using an algebraic notion (left coset).

2.1. Reduced BIBD

The simplest series of balanced incomplete block designs is unreduced BIBDs. These designs consist of all possible combinations of k out of v treatments. Balanced incomplete block designs exist for a wide range of design parameters. The basic design parameters are (v, b, k) which denote number of treatments, number of blocks and block size respectively. However, the only disadvantage it has is that in some cases, the size of experiment might be too large to work with in real life. When the number of treatments, v, is a perfect squares with v ≥ 9 and k < v the number of blocks required could be too large to cope with. This may force the experiment to adopt the reduced BIBDs. As such the reduced method is demonstrated in the construction of Designs 2, 3, 4 in section 4.3. Consequently, for instance, the design of unreduced BIBDs can be reduced by finding the highest common factor (H.C.F) of the parameters (b, r, λ) provided the value of H.C.F is neither decimal nor unity. The corresponding reduced designs could be obtained as: b¹ = b/f, r¹ = r/f, λ¹ = λ /f, where (b¹, r¹, λ¹) represents the parameters of the new (reduced) design and f is the H.C.F. of the parameters (b, r, λ). Example 3 and 4 below are to show the extent of reduction in the size of experiments that can be achieved with the use of reduced BIBD.

Example 3 Let us consider an unreduced design for case v =7 and k = 3, the design is given by the following 35 blocks:

The parameters of the design are:

v = 7, k = 3, b = 35, r = 15, λ = 5.

The corresponding reduced BIBD of example 3 is as presented as example 4 below:

Example 4 Reduced BIBD from unreduced BIBD in Example 3:

(1, 2, 4)

(1, 3, 7)

(1, 5, 6)

(2, 3, 5)

(2, 6, 7)

(3, 4, 5)

(4, 5, 7)

The parameters of the resulting reduced design are:

v = 7, k = 3, b¹ = 7, r¹ = 3, λ¹ = 1

which satisfies the necessary conditions for the existence of BIBDs.

2.2. Some Definitions, Basic Mathematical Concepts and Assumptions

Here, some mathematical terminologies, as well concepts, such as equivalence relations and partitions and some assumptions for the Coset-k² NBIBD of resolvable type, will be defined.

(a) Definition of Terms:

Two terms namely, left and right cost; and coset as Initial block will be defined in what follows:

Definition 2 (Left and Right Coset) Let G be a group, and H a subgroup of G, and g an element of G. Then

gH = {gh : h is an element of H} is called a left coset of H in G, by g and

Hg = {hg : h is an element of H} is called a right coset of H in G by g. If K = gh = Hg, then K is called a coset of H in G by g.

Definition 3 (Coset as Initial Block) Let H ≤ G. The quotient set G / H = {gH : g ϵ G} is called the Coset initial Block of H in G if G / H partitions G. A NBIBD of resolvable type with G / H as coset initial block is called a Coset-|H|² or Coset-|G| NBIBD of resolvable type.

(b) Concepts of Equivalence Relation and Partitions

Here, the concepts of Equivalence Relations and Partitions are presented because of their relevance in the proof of Theorem 5.

Let A and B be sets. Subsets of A x B are called relations. An equivalence relation on a set X is a relation R ϲ X × X such that

i. (x, x) ϵ R for all x ϵ X (reflexive property);

ii. (x, y) ϵ R implies (y, x) ϵ R (symmetric property);

iii. (x, y), (y, z) ϵ R imply (x, z) ϵ R (transitive property);

A partition P of a set X is a collection of non-empty sets X_i,

i ϵ Ω such that X_i ∩ X_j = ϕ for i ≠ j and.

Let R be an equivalence relation on a set X and let x ϵ X. Then [x] = {y ϵ X : yRx} is called the equivalence class of x.

Theorem 3 Fundamental Theorem of Equivalence Relation Given an equivalence relation R on a set X, the equivalence classes of X form a partition of X. Conversely, if P = {X_i}_iϵΩ is a partition of a set X, then there is an equivalence relation on X with equivalence classes X_i.

3. Theorems and Lemma for Construction of Designs

This section shall present a number of theorems to facilitate the design construction, and four assumptions that are to be compiled with are as follows:

1. v, number of treatments must be perfect square; and that v = k², where k is the size of the block;

2. if two treatments occur together in one sub-block, they must not occur together again in the subsequent sub-blocks throughout the construction of the designs;

3. number of main-blocks is k + 1; and

4. number of sub-blocks is k(k + 1)

Lemma 1 For a coset resolvable NBIBD with parameters satisfy the following relation.

(1)

The equality sign also holds if and only if BIBD is affine resolvable in each of the possible cases, as explained below.

Note that, if the conditions stated in equation (2) are satisfied, the required number of blocks for a Coset-k² resolvable NBIBD will be obtained.

3.1. Theorems on Coset-k² of NBIBD of Resolvable Type

Theorem 4 Let be a sub-block (generator) defined as H = {nk \ n = 1,…, k}, then i + H = {(i+h)(mod v)lh ϵ H}, i = 1, … k, with v( mod) = v, where k², is called a coset of H in ℕ by i ϵ N. The family {i + H}^k_i=1 is called the coset initial block of H in N. Let be the group of integers modulo k², with a subgroup H = {nk n = 1, 2,…, k}. Then

(i) is a coset of H in

(ii) is a Coset Initial Block of H in

Proof:

For k = 2,3,4 are illustrated below in cases 1,2 and 3 respectively.

The initial main-block for NBIBD for v = 4, k = 2, is [(3, 1), (4, 2)].

The initial main-block for NBIBD for v = 9, k = 3, is [(4, 7, 1), (5, 8, 2), (6, 9, 3)].

The initial main-block for NBIBD for v = 16, k = 4, is [(5, 9, 13, 1), (6, 10, 14, 2), (7, 11, 15, 3), (8, 12, 16, 4)].

Case 4: Let k = m, v = [1, 2, …, m²], H = {mn : n = 1, …, m} = {m, 2m, 3m, …, (m – 5)m, (m – 4)m, (m – 3)m, (m – 2)m, (m – 1)m, m²}. The sub-blocks which form the initial main-block are shown in Figure 1.

Case 5: Consider k = m + 1, v = [1, 2, …, (m + 1)²], H = {(m + 1)n : n = 1, …, m + 1} = {m + 1, 2(m + 1), 3(m + 1), …, (m - 1)(m + 1), m(m + 1), (m + 1)²}. The sub-blocks which form the initial main-block are shown in Figure 2.

The entries in the matrix in Figure 1 marked with a unique colour in diagonal lines correspond with entries in the matrix in Figure 2 marked with the same unique colour of horizontal lines in the appropriate direction of sequence. Entries in the last column of the matrix in Figure 1 correspond to the entries in the last column of matrix in Figure 2, except the entry in the last row; which corresponds to the entry in the row 1, column 1 of the matrix in Figure 1. All entries in the matrix of Figure 1 can be found in the matrix of Figure 2. However, the converse is not true because entries in the third and second to the last columns (except one entry ) of the matrix in Figure 2 are not in the matrix in the Figure 1. These entries are and All these entries are distinct and greater than Thus, when we have a Coset Initial Block of . So, when we equally have a Coset Initial Block of

Figure 1. Proof of Theorem 4 for k=m

Figure 2. Proof of Theorem 4 for k = m+1

Theorem 5: Let be a given treatment and let be partitions of X such that for the sub blocks Then is a Nested Balanced Incomplete Block Designs (NBIBD) of resolvable type with; treatment , sub-block size and parameter combinations

Proof:

Consider the pair where X = {1, 2, …, K²} and such that is a partition of X for each

Consider the pair where such that is a block design because X is a set of elements called points and is a collection of non-empty subsets of 2^X called blocks. Since

for each then is a complete block design with

Consider the pair where X = {1, 2, …, k²} and . is a block design because is a collection of non-empty subset , of X called sub-blocks. Recall that partitions X for each This implies for each Thus, for each i.e hence, is an Incomplete Block Design. Also, since for each appears once for That is the numbers of pair of treatments is one (λ = 1). Thus, is a BIBD with

Since partition X for each distinctly, then if then

Let be the equivalence relation corresponding to partition based on Theorem 3, then

Consider another partition of X. If then and so and i.e are not disjoint which is a contradiction. Thus whereas for any . Hence, the pair , appears once whereas . So, is a BIBD. Now, the Nested Block design is a Nested Balanced Incomplete Block Design (NBIBD) of a Resolvable type with parameters

Theorem 6: Let and let modulo k² with , i = 2, …, k+1 as partitions of such that for the blocks Then is a Coset-k² Nested Balanced Incomplete Block Designs (NBIBD) of Resolvable type with block size and treatment with parameters

Proof:

This is achieved by using Theorem 4 and Theorem 5. By Theorem 4, is a partition of X and so by Theorem 5, is a Nested Balanced Incomplete Block Designs (NBIBD) of Resolvable type with block size and treatment , With parameters

4. Construction of Designs

Here, nested balanced incomplete block designs for a number of parameter combinations will be constructed. In what immediately, appropriate model and relationship between design parameters shall be presented.

4.1. Model Specification

(2)

where, Y_ijl denotes the response from plot (unit) l in sub-block j of block i, μ an overall mean, β_i⁽¹⁾ the effect of the block i, β_ij⁽²⁾ the effect of the sub-block j in the block i, τ_ijl the effect of the treatment assigned to the unit (i, j, l) and ε_ijl a random error term, the error terms being assumed to be uncorrelated random variables with zero means and constant variance.

4.2. Relationship between Design Parameters

First, the design parameters are stated and defined as follow:

denotes the number of treatments, b number of blocks in the experiment, k size of each block (number of treatment per block), r number of replications for a given treatment in the experiment, λ number of times each pair of treatment appear (occur) together in the experiment, N total number of plots (observations), b₁ number of main-blocks in the experiment, b₂ number of sub-blocks in the experiment, k₁ size of each main-block (number of treatment per main-block), k₂ size of each sub-block (number of treatment per sub-block), λ₁ number of times each pair of treatment appear (occur) together in the main-blocks, λ₂ number of times each pair of treatment appear (occur) together in the sub-blocks and m number of sub-blocks within the main block.

Meanwhile, the relationships among the design parameters given above are presented below.

(3)

(4)

(5)

4.3. Nested BIBD’s for v = 4, 9, 16, and 25

Here, Theorem 4, Theorem 5, and Theorem 6 were continuously utilized for the construction of designs 1 to 6 for v = 4, v = 9 and v = 16 respectively.

Design 1: A resolvable NBIB Design v = 4, k = 2

[(3, 1),(4, 2)]

[(3, 4)(1, 2)]

[(1, 4),(2, 3)]

The parameters of the design are:

v = 4, r = 3, b₁ = 3, k₁ = 4, λ₁ = 3, b₂ = 6, k₂ = 2, λ₂ = 1

Design 2: A resolvable NBIB Design

v = 9, k = 3

The corresponding parameters of the reduced form are obtained as follows: Let f = gcd(b, r, λ) = 7, then

b¹ = b/f = 84/7 = 12, r¹ = r/f = 28/7 = 4, λ¹ = λ /f = 7/7 = 1

[(4, 7, 1), (5, 8, 2), (6, 9, 3)]

[(4, 5, 6), (7, 8, 9,),(1, 2, 3)]

[(1, 6, 8), (2, 4, 9), (3, 5, 7)]

[(1, 5, 9), (2, 6, 7), (3, 4, 8)]

The parameters of the design are:

V = 9, r = 4, b₁ = 4, k₁ = 9, λ₁ = 4, b₂ = 12, k₂ = 3, λ₂ = 1

Design 3: A resolvable NBIB design

v = 16, k = 4

The corresponding parameters of the reduced form are obtained as follows: Let f = gcd(b, r, λ) = 91, then

b¹ = b/f = 1820 / 91 = 20, r¹ = r / f = 455/91 = 5,

λ¹ = λ /f = 91/91 = 1

[(5, 9, 13, 1), (6, 10, 14, 2), (7, 11, 15, 3), (8,12 , 16, 4)]

[(5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16), (1, 2, 3, 4)]

[(1, 6, 11, 16), (2, 5, 12, 15), (3, 8, 9, 14), (4, 7, 10, 13)]

[(1, 7, 12, 14), (2, 8, 11, 13), (3, 5, 10, 16), (4, 6, 9, 15)]

[(1, 8, 10, 15), (2, 7, 9, 16), (3, 6, 12, 13), (4, 5, 11, 14)]

The parameters of the design are:

v = 16, r = 5, b₁ = 5, k₁ = 16, λ₁ = 5, b₂ = 20, k₂ = 4, λ₂ = 1

Design 4: A resolvable NBIB design v= 25, k = 5

The corresponding parameters of the reduced form are obtained as follows: Let f = gcd(b, r, λ) = 1771, then

b¹ = b/f = 53130/1771 = 30, r¹ = r/f = 10626/1771 = 6, λ¹ = λ /f = 1771/1771 = 1

[(6, 11, 16, 21, 1), (7, 12, 17, 22, 2), (8, 13, 18, 23, 3), (9, 14, 19, 24, 4), (10, 15, 20, 25, 5)]

[(6, 7, 8, 9, 10), (11, 12, 13, 14, 15), (16, 17, 18, 19, 20), (21, 22, 23, 24, 25), (1, 2, 3, 4, 5)]

[(1, 7, 13, 19, 25), (2, 10, 14, 16, 23), (3, 9, 12, 20, 21), (4, 6, 15, 18, 22), (5, 8, 11, 17, 24)]

[(1, 8, 14, 20, 22), (2, 9, 11, 18, 25), (3, 7, 15, 16, 24), (4, 10, 13, 17, 22), (5, 6, 12, 19, 23)]

[(1, 9, 15, 17, 23), (2, 6, 13, 20, 24), (3, 10, 11, 19, 22), (4, 8, 12, 16, 25), (5, 7, 14, 18, 21)]

[(1, 10, 12, 18, 24), (2, 8,15, 19, 21), (3, 6, 14, 17, 25), (4, 7, 11, 20, 23), (5, 9, 13, 16, 22)]

The parameters of the design are:

v = 25, r = 6, b₁ = 6, k₁ = 25, λ₁ = 6, b₂ = 30, k₂ = 5, λ₂ = 1

Design 5: A resolvable NBIB design v = 49, k = 7

Let f = gcd(b, r, λ) = 1533939, then

[(8, 15, 22, 29, 36, 43, 1), (9, 16, 23, 30, 37, 44, 2), (10. 17, 24, 31, 38, 45, 3), (11, 18, 25, 32, 39, 46, 4), (12, 19, 26, 33, 40, 47, 5), (13, 20, 27, 34, 41, 48, 6), (14, 21, 28, 35, 42, 49, 7)]

[(8, 9, 10, 11, 12, 13, 14), (15, 16, 17, 18, 19, 20, 21), (22, 23, 24, 25, 26, 27, 28), (29, 30, 31, 32, 33, 34, 35), (36, 37, 38, 39, 40, 41, 42), (43, 44, 45, 46, 47, 48, 49), (1, 2, 3, 4, 5, 6, 7)]

[(8, 16, 24, 32, 40, 48, 7), (15, 23, 31, 39, 47, 6, 14), (22, 30, 38, 46, 5, 13, 21), (29, 37, 45, 4, 12, 20, 28), (36, 44, 3, 11, 19, 27, 35), (43, 2, 10, 18, 26, 34, 42), (1, 9, 17, 25, 33, 41, 49)]

[(8, 23, 38, 4, 19, 34, 49), (15, 30, 45, 11, 26, 41, 7), (22, 37, 3, 18, 33, 48, 14), (29, 44, 10, 25, 40, 6, 21), (36, 2, 17, 32, 47, 13, 28), (43, 9, 24, 39, 5, 20, 35), (1, 16, 31, 46, 12, 27, 42)]

[(8, 30, 3, 25, 47, 20, 42), (15, 37, 10, 32, 5, 27, 49), (22, 44, 17, 39, 12, 34, 7), (29, 2, 24, 46, 19, 41, 14), (36, 9, 31, 4, 26, 48, 21), (43, 16, 38, 11, 33, 6, 28), (1, 23, 45, 18, 40, 13, 35)]

[(8, 37, 17, 46, 26, 6, 35), (15, 44, 24, 4, 33, 13, 42), (22, 2, 31, 11, 40, 20, 49), (29, 9, 38, 18, 47, 27, 7), (36, 16, 45, 25, 5, 34, 14), (43, 23, 3, 32, 12, 41, 21), (1, 30, 10, 39, 19, 48, 28)]

[(8, 44, 31, 18, 5, 41, 28), (15, 2, 38, 25, 12, 48, 35), (22, 9, 45, 32, 19, 6, 42), (29, 16, 3, 39, 26, 13, 49), (36, 23, 10, 46, 33, 20, 7), (43, 30, 17, 4, 40, 27, 14), (1, 37, 24, 11, 47, 34, 21)]

[(8, 2, 45, 39, 33, 27, 21), (15, 9, 3, 46, 40, 34, 28), (22, 16, 10, 4, 47, 41, 35), (29, 23, 17, 11, 5, 48, 42), (36, 30, 24, 18, 12, 6, 49), (43, 37, 31, 25, 19, 13, 7), (1, 44, 38, 32, 26, 20, 14)]

The parameters of the design are:

v = 49, r = 8, b₁ = 8, k₁ = 49, λ₁ = 8, b₂ = 56, k₂ = 7, λ₂ = 1

5. Conclusions

From all the designs constructed in section 4, it is indeed clear that they are unique designs because of the fact that; fewer number of blocks are required, even when designs with large number of treatments that are interest. Also the parameters of the design for any given number of treatments are specified with ease prior to the construction of the full designs. For any nested balanced incomplete block designs to be referred to as coset-k² resolvable, the design parameters are expected to satisfy the following relationship: v = k₁ = k₂², r = b₁ = λ₁, b₂ = b₁k₁ = rk₂ = λ₁k₂. The construction of designs when v is large requires tedious computational and combinatorial efforts.