Constant Block-Size with Constant Sum-Block Partially Balanced Incomplete Block Designs

Abstract

This chapter delves into the introduction, methodology, and application of Constant Block-Size with Constant Sum-Block Partially Balanced Incomplete Block Designs (CBS-CSB PBIBDs) as an innovative class of experimental designs. The study defines key concepts, including constant block-size, constant block-sum, and constant sum-block, and outlines conditions for implementing CBS-CSB PBIBDs. The research presents experimental designs for various scenarios, such as t = 21, k = 3; t = 15, k = 3; t = 27, k = 3, and illustrates their respective cases through line charts. Notably, the study demonstrates that as the number of replicates (r) increases, the efficiency factors also increase, and when r = 1, efficiency (E) equals 0. In conclusion, CBS-CSB PBIBDs emerge as a valuable tool for experimental prioritization, offering a new perspective on partially balanced incomplete block designs. The research highlights the significance of CBS-CSB PBIBDs across diverse fields of experimentation. The findings equip readers with the ability to define, design, and analyze experiments using CBS-CSB PBIBDs, while acknowledging challenges and limitations associated with this approach. This study contributes to the broader understanding of experimental design and provides a foundation for future research directions in this innovative field.

1. Introduction

Constant Block-Size with Constant Sum-Block Partially Balanced Incomplete Block Designs, CBS-CSB PBIBDs is a new class of partially balanced incomplete block designs that is use when it requires prioritizing the treatments combination. This new class of the design is significant in every aspect of experiment in physical, life, clinical, engineering, education, social science, psychology, pharmaceutical etc. once there is need to prioritize the treatment combination then CBS-CSB PBIBDs will be used.

The goal of this chapter is to define constant block-size partially balanced incomplete block designs and constant sum-block partially balanced incomplete block designs, design experiments using constant block-size with constant sum-block partially balanced incomplete block designs, CBS-CSB PBIBDs, state the Conditions for Constant Block-Size with Constant Sum-Block Partially Balanced Incomplete Block Designs, CBS-CSB PBIBDs, perform simple analysis on CBS-CSB PBIBDs and state some challenges and limitation on CBS-CSB PBIBDs.

In the remaining part of this chapter, we will discuss the background and theory, design principles and methodology, data analysis, discussion of results and conclusion of study’s findings.

1.1 Background and theory

The optimum design is balanced incomplete block designs (BIBDs) which gives 100% efficiency factor, e = 1. But the shortcoming of BIBDs leads to the introduction of partially balanced incomplete block designs (PBIBDs) introduced by the authors in [1]. This chapter introduces partially balanced incomplete block designs and its Efficiency Factors but its limitation was that it took into consideration the intra-block information only. The important properties of partially balanced incomplete block designs with two associate classes with both the intra-block and inter-block analysis in a comprehensive was introduced by the author [2]. Analysis of partially balanced incomplete block designs using simple square and rectangular lattices [3], construction of partially balanced incomplete block designs with two treatments per block and less or equal to ten (10) replications of each treatment [4], the class of partially balanced incomplete block designs with more than two associate classes comprehensively [5], m-classes partially balanced incomplete block designs and association scheme [6] and the table of 2-associate classes of partially balanced incomplete block designs were constructed [7].

There are several patterns and strategies in the construction of PBIBDs for the purpose of estimating higher efficiency factors. Some of the strategies are complicated and complex in designs and the efficiency factors is not satisfactory and comprehensive. The PBIBDs with higher associate classes were constructed by authors in [8, 9, 10]. Triangular and four associate classes PBIBDs with two replicates using dualization method were constructed by other scholars that contributed to the construction of PBIBDs with their different and respective pattern and strategies [11, 12, 13, 14]. In the physical and life experiment, there may be need to prioritize some treatment combinations. It was observed that development of PBIBDs through the incorporation of a constant block-sum strategy [15, 16] and later, in a subsequent work [17] delved deeper into and expanded upon this strategy in the construction of PBIBDs. This design changed the new pattern and strategy towards the construction of PBIBDs by authors cited in [18, 19, 20, 21, 22]. The limitations of the design was that r = 1, λ₁ = 1, λ₂ = 0. If r = 1, then E = 0 where ‘E’ is the efficiency factor, ‘r’ is the number of replicate and ‘λ’ is the number of treatment pair; this does not fit for analysis. To design an experiment, r $\ge$ 2, for r is positive integer. Instead, the design may be categorized into two or more to fit for analysis [23]. The key concepts for this work are Partially Balanced Incomplete Block Design; Constant Block-Size; Constant Sum-Block; Constant Block-Sum; Efficiency Factors.

2. Design principles and methodology

In this section, the paper defines, give conditions and design CBS-CSB PBIBDs.

2.1 Definition of terms

This sub-section defines Constant Block-Size, Constant Block-Sum and Constant Sum-Block of partially balanced incomplete block designs;

Definition 1: A design is said to have a constant block-size if the number of experimental unit, k, are the same irrespective of the number of blocks, b. For example;

• t = 21, k = 3, r = 1, b = 7

• t = 21, k = 3, r = 2, b = 14

• t = 21, k = 3, r = 3, b = 21

Definition 2: A design is said to be constant block-sum if the number of blocks, b, are the same, $\mathrm{b}=\frac{t}{k}$, and the sum of each corresponding number of experimental units are the same, irrespective of the number of treatments and replicates [24]. It is determined by the sum of all the treatments divided by number of blocks, $\mathrm{CBS}=\frac{\sum t}{b}$. Furthermore, Khattree model the design as:

t = pq, k = min (p,q), b = max (p,q), r = 1, λ₁ = 1, λ₂ = 0 where p and q must be odd numbers.

For example;

$$\mathrm{t}=21,\mathrm{k}=3,\mathrm{r}=1,\mathrm{b}=7,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=2,{\mathrm{n}}_2=18$$E1

$$\mathrm{CSB}=\frac{1+2+\dots +21}{7}=\frac{231}{7}=33.$$E2

Table 1 shows a Constant Block-Sum table, where each row (b1, b2, b3, etc.) represents a block, and the numbers within each block represent the treatments. The key feature of this table is that the sum of values in each block is constant and equal to 33.

					Sum
b1	1	11	21		33
b2	2	15	16		33
b3	3	10	20		33
b4	4	12	17		33
b5	5	9	19		33
b6	6	13	14		33
b7	7	8	18		33

Table 1.

Constant block-sum table.

Definition 3: A design is said to be a constant sum-block, if the following conditions exist;

• There is constant block size.

• Each corresponding blocks has constant sum.

• The number of blocks are not the same, $\mathrm{b}=\frac{\mathit{tr}}{k}$

• The number of treatments, t, are constant.

• The number of replicates, r, are at least one, $\mathrm{r}\ge \left(\mathrm{1}\right)$

For example;

$$\mathrm{t}=21,\mathrm{k}=3,\mathrm{r}=2,\mathrm{b}=14,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=4,{\mathrm{n}}_2=16.$$

In Table 2, each row represents a block labeled as b1, b2, b3, and so on. The numbers within each block represent different treatments. For example, in block b1, the treatments are 1, 11, 21, and the sum of these treatments is 33. The same pattern follows for the other blocks. The sum of treatments in each block is the same and equal to 33. This is evident by the “Sum” column at the end of each row, indicating that the values in each row (treatments within a block) add up to 33. This design, with a constant sum within each block is used in experimental design to ensure balance and control for variability. It allows for a systematic distribution of treatments across different blocks while maintaining a consistent sum within each block.

					Sum
b1	1	11	21		33
b2	2	15	16		33
b3	3	10	20		33
b4	4	12	17		33
b5	5	9	19		33
b6	6	13	14		33
b7	7	8	18		33
b8	1	12	20		33
b9	2	14	17		33
b10	3	9	21		33
b11	4	11	18		33
b12	5	13	15		33
b13	6	8	19		33
b14	7	10	16		33

Table 2.

Constant sum-block table.

2.2 Conditions for constant block-size with constant sum-block partially balanced incomplete block designs, CBS-CSB PBIBDs

There are conditions for CBS-CSB PBIBDs;

• The treatments combinations must be prioritized, that is, there must be number of zero associates, λ₂ = 0.

• The maximum number of replicates must be necessarily determine from treatment one.

• The number of treatments and block sizes remain fixed.

For example;

The designs;

$$\mathrm{t}=21,\mathrm{k}=3,{\mathrm{\lambda }}_2=0,$$

The three numerical values in each row in Table 3 represent different treatments or conditions. For example, in the first row (SN = 1), the values are 1, 21, and 11 and their sum is 33. Similarly, the sum of values in each subsequent row is consistently 33. The key feature here, as in the previous tables, is that the sum of the numerical values in each row is constant and equal to 33. This ensures a balanced distribution of the treatments or conditions. In this context, the table structure represents a specific design where the sum of values in each row is held constant. The associate for treatment 1 is given in Table 4.

SN					Sum
1	1	21	11		33
2	1	20	12		33
3	1	19	13		33
4	1	18	14		33
5	1	17	15		33

Table 3.

Maximum number of replicates table.

T	λ1 = 1	λ2 = 0
1	11, 21,12,20,13,19,14,18,15,17	2,3,4,5,6,7,8,9,10,16

Table 4.

Associate table for treatment 1.

Table 4 associates different sets of values with two conditions, denoted as λ₁ and λ₂, for a specific treatment labeled as T. The row labeled as T corresponds to a specific treatment condition while λ1 and λ2 are conditions or parameters associated with the treatment. For treatment 1, the associated values are presented under λ1 = 1 and λ2 = 0. Under each condition (λ1 and λ2), there are sets of values associated with the treatment. For example, under λ1 = 1, the associated values are 11, 21, 12, 20, 13, 19, 14, 18, 15, 17. Under λ2 = 0, the associated values are 2, 3, 4, 5, 6, 7, 8, 9, 10, 16. Table 4 shows that 1 < r $\le$ 5 and λ₂ = 0. This met the conditions for CBS-CSB PBIBDs.

2.3 Case studies on the design constant block-size with constant sum-block partially balanced incomplete block designs, CBS-CSB PBIBDs

This section shows the construction of CBS-CSB PBIBDs. The researcher and the user will determine the number of replicates based on the priority of the treatment combination and the cost of the experiment (Table 5). This means $1<\mathrm{r}\le max\left(\mathrm{r}\right)$.

					Sum
b1	1	11	21		33
b2	2	15	16		33
b3	3	10	20		33
b4	4	12	17		33
b5	5	9	19		33
b6	6	13	14		33
b7	7	8	18		33

Table 5.

Case 1 design.

2.3.1 Construction of design 1

The design t = 21, k = 3, λ₂ = 0 will be constructed in five cases since r = 5.

Case 1: t = 21, k = 3, r = 1, b = 7, λ₁ = 1, λ₂ = 0, n₁ = 2, n₂ = 18.

The table represents the design with 7 blocks (b1 to b7), and each block contains 3 treatments (k = 3). The values in each block are arranged in a way that the sum of treatments within each block is constant and equal to 33. This design follows a structure for the Constant Block-Sum table provided earlier, with the same sum of treatments in each block.

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 0\\ 0& 18\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}0& 2\\ 2& 15.\end{array}\right]$$

$${\mathrm{E}}_1=0,{\mathrm{E}}_2=0,\mathrm{E}=0$$

The efficiency factor (E) is calculated using R software based on a method proposed by authors in [25]. Table 6 associates each treatment (T) with sets of values for the 1st and 2nd conditions. For example, treatment 1 (T = 1) is associated with values 11 and 21 under the 1st condition, and with values 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20 under the 2nd condition. The structure of the table indicates how each treatment is linked to specific sets of values under different conditions. This indicates that the efficiency is not directly given but needs to be calculated using specific software and methodology. Case 2 designs are shown in Tables 7 and 8, respectively.

T	1st	2nd
1	11, 21	2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20
2	15,16	1,3,4,5,6,7,8,9,10,11,12,13,14,17,18,19,20,21
3	10, 20	1,2,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21
4	12, 17	1,2,3,5,6,7,8,9,10,11,13,14,15,16,18,19,20,21
5	9, 19	1,2,3,4,6,7,8,10,11,12,13,14,15,16,17,18,20,21
6	13, 14	1,2,3,4,5,7,8,9,10,11,12,15,16,17,18,19,20,21
7	8, 18	1,2,3,4,5,6,9,10,11,12,13,14,15,16,17,19,20,21
8	7, 18	1,2,3,4,5,6,9,10,11,12,13,14,15,16,17,19,20,21
9	5, 19	1,2,3,4,6,7,8,10,11,12,13,14,15,16,17,18,20,21
10	3, 20	1,2,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21
11	1, 21	2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20
12	4,17	1,2,3,5,6,7,8,9,10,11,13,14,15,16,18,19,20,21
13	6,14	1,2,3,4,5,7,8,9,10,11,12,15,16,17,18,19,20,21
14	6,13	1,2,3,4,5,7,8,9,10,11,12,15,16,17,18,19,20,21
15	2,16	1,3,4,5,6,7,8,9,10,11,12,13,14,17,18,19,20,21
16	2,15	1,3,4,5,6,7,8,9,10,11,12,13,14,17,18,19,20,21
17	4,12	1,2,3,5,6,7,8,9,10,11,13,14,15,16,18,19,20,21
18	7,8	1,2,3,4,5,6,9,10,11,12,13,14,15,16,17,19,20,21
19	5,9	1,2,3,4,6,7,8,10,11,12,13,14,15,16,17,18,20,21
20	3,10	1,2,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21
21	1,11	2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20

Table 6.

Case 1 associates.

					Sum
b1	1	11	21		33
b2	2	15	16		33
b3	3	10	20		33
b4	4	12	17		33
b5	5	9	19		33
b6	6	13	14		33
b7	7	8	18		33
b8	1	12	20		33
b9	2	14	17		33
b10	3	9	21		33
b11	4	11	18		33
b12	5	13	15		33
b13	6	8	19		33
b14	7	10	16		33

Table 7.

Case 2 designs.

T	1st	2nd
1	11, 21,12,20	2,3,4,5,6,7,8,9,10,13,14,15,16,17,18,19
2	15,16,14,17	1,3,4,5,6,7,8,9,10,11,12,13,18,19,20,21
3	10, 20,9,21	1,2,4,5,6,7,8,11,12,13,14,15,16,17,18,19
4	12, 17,11,18	1,2,3,5,6,7,8,9,10,13,14,15,16,19,20,21
5	9, 19,13,15	1,2,3,4,6,7,8,10,11,12,14,16,17,18,20,21
6	13, 14,8,19	1,2,3,4,5,7,9,10,11,12,15,16,17,18,20,21
7	8, 18,10,16	1,2,3,4,5,6,9,11,12,13,14,15,17,19,20,21
8	7, 18,6,19	1,2,3,4,5,9,10,11,12,13,14,15,16,17,20,21
9	5, 19,3,21	1,2,4,6,7,8,10,11,12,13,14,15,16,17,18,20
10	3, 20,7,16	1,2,4,5,6,8,9,11,12,13,14,15,17,18,19,21
11	1, 21,4,18	2,3,5,6,7,8,9,10,12,13,14,15,16,17,19,20
12	4,17,1,20	2,3,5,6,7,8,9,10,11,13,14,15,16,18,19,21
13	6,14,5,15	1,2,3,4,7,8,9,10,11,12,16,17,18,19,20,21
14	6,13,2,17	1,3,4,5,7,8,9,10,11,12,15,16,18,19,20,21
15	2,16,5,13	1,3,4,6,7,8,9,10,11,12,14,17,18,19,20,21
16	2,15,7,10	1,3,4,5,6,8,9,11,12,13,14,17,18,19,20,21
17	4,12,2,14	1,3,5,6,7,8,9,10,11,13,15,16,18,19,20,21
18	7,8,4,11	1,2,3,5,6,9,10,12,13,14,15,16,17,19,20,21
19	5,9,6,8	1,2,3,4,7,10,11,12,13,14,15,16,17,18,20,21
20	3,10,1,12	2,4,5,6,7,8,9,11,13,14,15,16,17,18,19,21
21	1,11,3,9	2,4,5,6,7,8,10,12,13,14,15,16,17,18,19,20

Table 8.

Case 2 associates.

Case 2: t = 21, k = 3, r = 2, b = 14, λ₁ = 1, λ₂ = 0, n₁ = 4, n₂ = 16

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 2\\ 2& 14\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}0& 4\\ 4& 11\end{array}\right]$$

$${\mathrm{E}}_1=0.6667,{\mathrm{E}}_2=0.4444,\mathrm{E}=0.4762$$

Each row in Table 7 represents a block, and there are 14 blocks in total. The numbers within each block represent different treatments. The sum of treatments in each block is constant and equal to 33, maintaining a similar structure to the design in Case 1. Table 8 associates each treatment (T) with sets of values for the 1st and 2nd conditions. For instance, treatment 1 (T = 1) is associated with values 11, 21, 12, and 20 under the 1st condition, and with values 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19 under the 2nd condition.

Case 3: t = 21, k = 3, r = 3, b = 21, λ₁ = 1, λ₂ = 0, n₁ = 6, n₂ = 14

$${\mathrm{P}}_1=\left[\begin{array}{cc}2& 3\\ 3& 11\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}2& 4\\ 4& 9\end{array}\right]$$

$${\mathrm{E}}_1=0.7111,{\mathrm{E}}_2=0.5926,\mathrm{E}=0.6238$$

The table represents the design with 21 blocks (b1 to b21), and each block contains 3 treatments (k = 3). The values in each block are arranged in a way that the sum of treatments within each block is constant and equal to 33. This design follows a structure similar to the Constant Block-Sum tables provided earlier, with the same sum of treatments in each block (Table 9). Table 10 is similar to the previous associates tables. It associates each treatment (T) with sets of values for the 1st and 2nd conditions. For example, treatment 1 (T = 1) is associated with values 11, 21, 12, 20, 13, and 19 under the 1st condition, and with values 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18 under the 2nd condition.

					Sum
b1	1	11	21		33
b2	2	15	16		33
b3	3	10	20		33
b4	4	12	17		33
b5	5	9	19		33
b6	6	13	14		33
b7	7	8	18		33
b8	1	12	20		33
b9	2	14	17		33
b10	3	9	21		33
b11	4	11	18		33
b12	5	13	15		33
b13	6	8	19		33
b14	7	10	16		33
b15	1	13	19		33
b16	2	11	20		33
b17	3	14	16		33
b18	4	8	21		33
b19	5	10	18		33
b20	6	12	15		33
b21	7	9	17		33

Table 9.

Case 3 design.

T	1st	2nd
1	11, 21,12,20,13,19	2,3,4,5,6,7,8,9,10,14,15,16,17,18
2	15,16,14,17,11,20	1,3,4,5,6,7,8,9,10,12,13,18,19,21
3	10, 20,9,21,14,16	1,2,4,5,6,7,8,11,12,13,15,17,18,19
4	12, 17,11,18,8,21	1,2,3,5,6,7,9,10,13,14,15,16,19,20
5	9, 19,13,15,10,18	1,2,3,4,6,7,8,11,12,14,16,17,20,21
6	13, 14,8,19,12,15	1,2,3,4,5,7,9,10,11,16,17,18,20,21
7	8, 18,10,16,9,17	1,2,3,4,5,6,11,12,13,14,15,19,20,21
8	7, 18,6,19,4,21	1,2,3,5,9,10,11,12,13,14,15,16,17,20
9	5, 19,3,21,7,17	1,2,4,6,8,10,11,12,13,14,15,16,18,20
10	3, 20,7,16,5,18	1,2,4,6,8,9,11,12,13,14,15,17,19,21
11	1, 21,4,18,2,20	3,5,6,7,8,9,10,12,13,14,15,16,17,19
12	4,17,1,20,6,15	2,3,5,7,8,9,10,11,13,14,16,18,19,21
13	6,14,5,15,1,19	2,3,4,7,8,9,10,11,12,16,17,18,20,21
14	6,13,2,17,3,16	1,4,5,7,8,9,10,11,12,15,18,19,20,21
15	2,16,5,13,6,12	1,3,4,7,8,9,10,11,14,17,18,19,20,21
16	2,15,7,10,3,14	1,4,5,6,8,9,11,12,13,17,18,19,20,21
17	4,12,2,14,7,9	1,3,5,6,8,10,11,13,15,16,18,19,20,21
18	7,8,4,11,5,10	1,2,3,6,9,12,13,14,15,16,17,19,20,21
19	5,9,6,8,1,13	2,3,4,7,10,11,12,14,15,16,17,18,20,21
20	3,10,1,12,2,11	4,5,6,7,8,9,13,14,15,16,17,18,19,21
21	1,11,3,9,4,8	2,5,6,7,10,12,13,14,15,16,17,18,19,20

Table 10.

Case 3 associates.

Case 4: t = 21, k = 3, r = 4, b = 21, λ₁ = 1, λ₂ = 0, n₁ = 8, n₂ = 12

$${\mathrm{P}}_1=\left[\begin{array}{cc}3& 4\\ 4& 8\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}5& 3\\ 3& 8\end{array}\right]$$

$${\mathrm{E}}_1=0.713,{\mathrm{E}}_2=0.6417,\mathrm{E}=0.6684$$

Table 11 represents the design for Case 4, with specific values assigned to each block. Each row corresponds to a block (b1 to b28), and within each block, there are different treatments arranged in a way that the sum of treatments is constant and equal to 33. Table 12 is similar to the previous associate tables. It associates each treatment (T) with sets of values for the 1st and 2nd conditions. For example, treatment 1 (T = 1) is associated with values 11, 21, 12, 20, 13, 19, 14, and 18 under the 1st condition, and with values 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17 under the 2nd condition. The notations P1 and P2 are partitions with specific characteristics. The E1 = 0.713, E2 = 0.6417, E = 0.6684 values represent efficiency factors. In experimental design, efficiency is a measure of how well an experimental design utilizes the available resources. Here, E1 and E2 are specific efficiency values for P1 and P2, and E is an overall efficiency factor.

					Sum
b1	1	11	21		33
b2	2	15	16		33
b3	3	10	20		33
b4	4	12	17		33
b5	5	9	19		33
b6	6	13	14		33
b7	7	8	18		33
b8	1	12	20		33
b9	2	14	17		33
b10	3	9	21		33
b11	4	11	18		33
b12	5	13	15		33
b13	6	8	19		33
b14	7	10	16		33
b15	1	13	19		33
b16	2	11	20		33
b17	3	14	16		33
b18	4	8	21		33
b19	5	10	18		33
b20	6	12	15		33
b21	7	9	17		33
b22	1	14	18		33
b23	2	12	19		33
b24	3	13	17		33
b25	4	9	20		33
b26	5	7	21		33
b27	6	11	16		33
b28	8	10	15		33

Table 11.

Case 4 design.

T	1st	2nd
1	11, 21,12,20,13,19,14,18	2,3,4,5,6,7,8,9,10,15,16,17
2	15,16,14,17,11,20,12,19	1,3,4,5,6,7,8,9,10,13,18,21
3	10, 20,9,21,14,16,13,17	1,2,4,5,6,7,8,11,13,15,18,19
4	12, 17,11,18,8,21,9,20	1,2,3,5,6,7,10,13,14,15,16,19
5	9, 19,13,15,10,18,7,21	1,2,3,4,6,8,11,12,14,16,17,20
6	13, 14,8,19,12,15,11,16	1,2,3,4,5,7,9,10,17,18,20,21
7	8, 18,10,16,9,17,5,21	1,2,3,4,6,11,12,13,14,15,19,20
8	7, 18,6,19,4,21,10,15	1,2,3,5,9,11,12,13,14,16,17,20
9	5, 19,3,21,7,17,4,20	1,2,6,8,10,11,12,13,14,15,16,18
10	3, 20,7,16,5,18,8,15	1,2,4,6,9,11,12,13,14,17,19,21
11	1, 21,4,18,2,20,6,16	3,5,7,8,9,10,12,13,14,15,17,19
12	4,17,1,20,6,15,2,19	3,5,7,8,9,10,11,13,14,16,18,21
13	6,14,5,15,1,19,3,17	2,4,7,8,9,10,11,12,16,18,20,21
14	6,13,2,17,3,16,1,18	4,5,7,8,9,10,11,12,15,19,20,21
15	2,16,5,13,6,12,8,10	1,3,4,7,9,11,14,17,18,19,20,21
16	2,15,7,10,3,14,6,11	1,4,5,8,9,12,13,17,18,19,20,21
17	4,12,2,14,7,9,3,13	1,5,6,8,10,11,15,16,18,19,20,21
18	7,8,4,11,5,10,1,14	2,3,6,9,12,13,15,16,17,19,20,21
19	5,9,6,8,1,13,2,12	3,4,7,10,11,14,15,16,17,18,20,21
20	3,10,1,12,2,11,4,9	5,6,7,8,13,14,15,16,17,18,19,21
21	1,11,3,9,4,8,5,7	2,6,10,12,13,14,15,16,17,18,19,20

Table 12.

Case 4 associates.

Case 5: t = 21, k = 3, r = 5, b = 21, λ₁ = 1, λ₂ = 0, n₁ = 10, n₂ = 10

$${\mathrm{P}}_1=\left[\begin{array}{cc}4& 5\\ 5& 5\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}8& 2\\ 2& 7\end{array}\right]$$

$${\mathrm{E}}_1=0.7077,{\mathrm{E}}_2=0.6571,\mathrm{E}=0.6815$$

Table 13 presents a Case 5 design. It represents the design for Case 5, with specific values assigned to each block. Each row corresponds to a block (b1 to b35), and within each block, there are different treatments arranged in a way that the sum of treatments is constant and equal to 33. Table 14 contained Case 5 associates. This table shows the associations between the treatments in Case 5. Each row represents a treatment (T1 to T21), and the “1st” and “2nd” columns indicate which other treatments are associated with the corresponding treatment. For example, the first row indicates that for Treatment 1 (T1), the associated treatments in the 1st group are 11, 21, 12, 20, 13, 19, 14, 18, 15, 17, and in the 2nd group are 2, 3, 4, 5, 6, 7, 8, 9, 10, 16. Similarly, each subsequent row provides information about the associations for the corresponding treatment. The efficiency values (E1, E2, and E) are also provided as 0.7077, 0.6571, and 0.6815, respectively (Figure 1).

					Sum
b1	1	11	21		33
b2	2	15	16		33
b3	3	10	20		33
b4	4	12	17		33
b5	5	9	19		33
b6	6	13	14		33
b7	7	8	18		33
b8	1	12	20		33
b9	2	14	17		33
b10	3	9	21		33
b11	4	11	18		33
b12	5	13	15		33
b13	6	8	19		33
b14	7	10	16		33
b15	1	13	19		33
b16	2	11	20		33
b17	3	14	16		33
b18	4	8	21		33
b19	5	10	18		33
b20	6	12	15		33
b21	7	9	17		33
b22	1	14	18		33
b23	2	12	19		33
b24	3	13	17		33
b25	4	9	20		33
b26	5	7	21		33
b27	6	11	16		33
b28	8	10	15		33
b29	1	15	17		33
b30	2	10	21		33
b31	3	11	19		33
b32	4	13	16		33
b33	5	8	20		33
b34	6	9	18		33
b35	7	12	14		33

Table 13.

Case 5 design.

T	1st	2nd
1	11, 21,12,20,13,19,14,18,15,17	2,3,4,5,6,7,8,9,10,16
2	15,16,14,17,11,20,12,19,10,21	1,3,4,5,6,7,8,9,13,18
3	10, 20,9,21,14,16,13,17,11,19	1,2,4,5,6,7,8,13,15,18
4	12, 17,11,18,8,21,9,20,13,16	1,2,3,5,6,7,10,14,15,19
5	9, 19,13,15,10,18,7,21,8,20	1,2,3,4,6,11,12,14,16,17
6	13, 14,8,19,12,15,11,16,9,18	1,2,3,4,5,7,10,17,20,21
7	8, 18,10,16,9,17,5,21,12,14	1,2,3,4,6,11,13,15,19,20
8	7, 18,6,19,4,21,10,15,5,20	1,2,3,9,11,12,13,14,16,17
9	5, 19,3,21,7,17,4,20,6,18	1,2,8,10,11,12,13,14,15,16
10	3, 20,7,16,5,18,8,15,2,21	1,4,6,9,11,12,13,14,17,19
11	1, 21,4,18,2,20,6,16,3,19	5,7,8,9,10,12,13,14,15,17
12	4,17,1,20,6,15,2,19,7,14	3,5,8,9,10,11,13,16,18,21
13	6,14,5,15,1,19,3,17,4,16	2,7,8,9,10,11,12,18,20,21
14	6,13,2,17,3,16,1,18,7,12	4,5,8,9,10,11,15,19,20,21
15	2,16,5,13,6,12,8,10,1,17	3,4,7,9,11,14,18,19,20,21
16	2,15,7,10,3,14,6,11,4,13	1,5,8,9,12,17,18,19,20,21
17	4,12,2,14,7,9,3,13,1,15	5,6,8,10,11,16,18,19,20,21
18	7,8,4,11,5,10,1,14,6,9	2,3,12,13,15,16,17,19,20,21
19	5,9,6,8,1,13,2,12,3,11	4,7,10,14,15,16,17,18,20,21
20	3,10,1,12,2,11,4,9,5,8	6,7,13,14,15,16,17,18,19,21
21	1,11,3,9,4,8,5,7,2,10	6,12,13,14,15,16,17,18,19,20

Table 14.

Case 5 associates.

Figure 1.

Design parameters (t = 21, k = 3).

2.3.2 Construction of design 2

The design t = 15, k = 3, λ₂ = 0 will be constructed in five cases since r = 4.

Case 1: t = 15, k = 3, r = 1, b = 10, λ₁ = 1, λ₂ = 0, n₁ = 2, n₂ = 12

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 0\\ 0& 12\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}0& 2\\ 2& 9\end{array}\right]$$

$${\mathrm{E}}_1=0,{\mathrm{E}}_2=0,\mathrm{E}=0$$

Case 2

$$\mathrm{t}=15,\mathrm{k}=3,\mathrm{r}=2,\mathrm{b}=10,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=4,{\mathrm{n}}_2=10$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 2\\ 2& 8\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}2& 2\\ 2& 7\end{array}\right]$$

$${\mathrm{E}}_1=0.75,{\mathrm{E}}_2=0.6,\mathrm{E}=0.6364$$

Case 3

$$\mathrm{t}=15,\mathrm{k}=3,\mathrm{r}=3,\mathrm{b}=15,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=6,{\mathrm{n}}_2=8$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}2& 3\\ 3& 5\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}4& 2\\ 2& 5\end{array}\right]$$

$${\mathrm{E}}_1=0.7302,{\mathrm{E}}_2=0.6389,\mathrm{E}=0.6751$$

Case 4

$$\mathrm{t}=15,\mathrm{k}=3,\mathrm{r}=4,\mathrm{b}=20,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=8,{\mathrm{n}}_2=6$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}4& 3\\ 3& 3\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}7& 1\\ 1& 4\end{array}\right]$$

$${\mathrm{E}}_1=0.725,{\mathrm{E}}_2=0.6591,\mathrm{E}=0.6952$$

For Case 1 an efficiency of E = 0 implies a 100% efficient design. By achieving E1 = E2 = E = 0, the allocations defined in the P1 and P2 matrices result in a maximally efficient balanced incomplete block design for estimating the main effects of the 3 factors over the 15 runs. Case 2 defines a constant block-size with constant sum-block partially balanced incomplete block (PBIB) design. This achieves variances and overall design efficiency: E1 = 0.75 E2 = 0.6 E = 0.6364. An efficiency of 0.6364 implies the partial blocking and subsampling still allows reasonably efficient estimation of the main effects for the 3 factors at strength 2. The allocations balance runs across subsets to optimize estimation of 2-factor interactions at the cost of some design efficiency compared to a fully balanced design (E = 0).

Case 3 also defines a constant block-size with constant sum-block partially balanced incomplete block (PBIB) design. This allocation achieves variances and efficiency: E1 = 0.7302, E2 = 0.6389 and E = 0.6751. The partial blocking balances runs across subsets to allow estimation of 3-factor interactions. An efficiency of 0.6751 indicates the subsample allocations still allow moderately efficient estimation of main effects, although less precisely than Case 1 or 2 constructions. The matrices P1 and P2 systematically distribute runs to balance representation across subsets at the cost of some variance inflation. In Case 4, the allocation of runs to the two subsamples allows estimation of 3-factor interactions between the variables. The partial blocking distributes runs systematically though unequally across subsets. This achieves reasonable variance for assessing main effects, traded off against balance. Appropriate analysis can still estimate factor impacts. The matrices P1 and P2 define the subdivision rules that split the 15 runs to optimize estimation of 3-way interactions between the 3 factors. The efficiency factor, E, increases as the number of case increases (Figures 2 and 3).

Figure 2.

Design parameters (t = 15, k = 3).

Figure 3.

Design parameters (t = 27, k = 3).

2.3.3 Construction of design 3

The design t = 27, k = 3, λ₂ = 0 will be constructed in five cases since r = 7.

Case 1

$$\mathrm{t}=27,\mathrm{k}=3,\mathrm{r}=1,\mathrm{b}=9,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=2,{\mathrm{n}}_2=24$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 0\\ 0& 24\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}0& 2\\ 2& 21\end{array}\right]$$

$${\mathrm{E}}_1=0,{\mathrm{E}}_2=0,{\mathrm{E}}_3=0$$

Case 2

$$\mathrm{t}=27,\mathrm{k}=3,\mathrm{r}=2,\mathrm{b}=18,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=4,{\mathrm{n}}_2=22$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 2\\ 2& 20\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}3& 1\\ 1& 20\end{array}\right]$$

$${\mathrm{E}}_1=0.7667,{\mathrm{E}}_2=0.6389,{\mathrm{E}}_3=0.6557$$

Case 3

$$\mathrm{t}=27,\mathrm{k}=3,\mathrm{r}=3,\mathrm{b}=27,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=6,{\mathrm{n}}_2=20$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 4\\ 4& 16\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}5& 1\\ 1& 18\end{array}\right]$$

$${\mathrm{E}}_1=0.7284,{\mathrm{E}}_2=0.6556,{\mathrm{E}}_3=0.671$$

Case 4

$$\mathrm{t}=27,\mathrm{k}=3,\mathrm{r}=4,\mathrm{b}=36,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=8,{\mathrm{n}}_2=18$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 6\\ 6& 12\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}7& 1\\ 1& 16\end{array}\right]$$

$${\mathrm{E}}_1=0.7115,{\mathrm{E}}_2=0.6607,{\mathrm{E}}_3=0.6756$$

Case 5

$$\mathrm{t}=27,\mathrm{k}=3,\mathrm{r}=5,\mathrm{b}=45,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=10,{\mathrm{n}}_2=16$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 8\\ 8& 8\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}9& 1\\ 1& 14\end{array}\right]$$

$${\mathrm{E}}_1=0.702,{\mathrm{E}}_2=0.663,{\mathrm{E}}_3=0.6774$$

Case 6

$$\mathrm{t}=27,\mathrm{k}=3,\mathrm{r}=6,\mathrm{b}=54,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=12,{\mathrm{n}}_2=14$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 10\\ 10& 4\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}11& 1\\ 1& 12\end{array}\right]$$

$${\mathrm{E}}_1=0.6958,{\mathrm{E}}_2=0.6641,{\mathrm{E}}_3=0.6784$$

Case 7

$$\mathrm{t}=27,\mathrm{k}=3,\mathrm{r}=7,\mathrm{b}=63,{\mathrm{\lambda }}_1=1,{\mathrm{\lambda }}_2=0,{\mathrm{n}}_1=14,{\mathrm{n}}_2=12$$

$${\mathrm{P}}_1=\left[\begin{array}{cc}1& 11\\ 11& 2\end{array}\right],{\mathrm{P}}_2=\left[\begin{array}{cc}13& 1\\ 1& 10\end{array}\right]$$

$${\mathrm{E}}_1=0.6914,{\mathrm{E}}_2=0.6648,{\mathrm{E}}_3=0.6789$$

3. Data analysis

3.1 Dataset description

This dataset was collected on the impact of Body Mass Index (BMI) on impact oscillometry (IOS) measures on children with sickle cell diseases. The source of the dataset [26].

For the purpose of this chapter, 45 observations were extracted which has the following factors; asthma, hydroxyurea, ICS and LABA. Furthermore, each treatment was replicated 3 times alongside with the gender. The BMI results were recorded.

Table 15 dataset will be addressed in two cases (2 and 3) based on the priorities. The priority is based on the fixed sum of 24 (Tables 16–21).

SN	Subject ID	Hydroxyurea	Asthma	ICS	LABA	REP	Gender	BMI
1	1	Yes	Yes	Yes	Yes	1	Male	23.6
2	1	Yes	Yes	Yes	Yes	1	Male	23.74
3	8	Yes	Yes	Yes	Yes	1	Male	18.39
4	1	Yes	Yes	Yes	No	2	Male	22.84
5	1	Yes	Yes	Yes	No	2	Male	23.53
6	3	Yes	Yes	Yes	No	2	Male	20.95
7	3	Yes	Yes	No	No	3	Male	20.79
8	21	Yes	Yes	No	No	3	Male	15.87
9	30	Yes	Yes	No	No	3	Male	13.98
10	13	Yes	No	No	No	4	Male	16.36
11	13	Yes	No	No	No	4	Male	17.52
12	13	Yes	No	No	No	4	Male	19.07
13	25	No	Yes	Yes	Yes	5	Male	17.68
14	59	No	Yes	Yes	Yes	5	male	16.71
15	64	No	Yes	Yes	Yes	5	male	16.01
16	24	No	Yes	Yes	No	6	Male	15.36
17	25	No	Yes	Yes	No	6	Male	16.03
18	25	No	Yes	Yes	No	6	Male	16.26
19	24	No	Yes	No	No	7	Male	15.25
20	37	No	Yes	No	No	7	Male	15.09
21	37	No	Yes	No	No	7	Male	15.83
22	15	No	No	No	No	8	Male	22.03
23	36	No	No	No	No	8	Male	14.81
24	36	No	No	No	No	8	Male	14.83
25	10	Yes	Yes	Yes	Yes	9	Female	20.19
26	10	Yes	Yes	Yes	Yes	9	Female	20.49
27	10	Yes	Yes	Yes	Yes	9	Female	21.66
28	14	Yes	Yes	No	No	10	Female	16.02
29	14	Yes	Yes	No	No	10	Female	16.79
30	14	Yes	Yes	No	No	10	Female	17.8
31	57	No	Yes	Yes	Yes	11	Female	17.68
32	57	No	Yes	Yes	Yes	11	Female	16.82
33	57	No	Yes	Yes	Yes	11	Female	16.75
34	2	No	No	No	No	12	Female	27.82
35	4	No	No	No	No	12	Female	22.29
36	4	No	No	No	No	12	Female	22.35
37	12	Yes	Yes	Yes	No	13	Female	16.41
38	12	Yes	Yes	Yes	No	13	Female	17.39
39	12	Yes	Yes	Yes	No	13	Female	18.16
40	11	Yes	No	No	No	14	Female	16.13
41	11	Yes	No	No	No	14	Female	17.26
42	11	Yes	No	No	No	14	Female	21
43	7	No	Yes	Yes	No	15	Female	20.98
44	16	No	Yes	Yes	No	15	Female	20.8
45	44	No	Yes	Yes	No	15	Female	14.93

Table 15.

Table showing the impact of BMI on IOS.

					Sum
b1	1(23.6)	8(22.03)	15(20.98)		24
b2	1(23.74)	11(17.68)	12(27.82)		24
b3	2(22.84)	10(16.02)	12(22.29)		24
b4	2(23.53)	7(15.25)	15(20.8)		24
b5	3(20.79)	7(15.09)	14(16.13)		24
b6	315.87)	8(14.81)	13(16.41)		24
b7	4(16.36)	9(20.19)	11(16.82)		24
b8	4(17.52)	6(15.36)	14(17.26)		24
b9	5(17.68)	6(16.03)	13(17.39)		24
b10	5(16.71)	9(20.49)	10(16.79)		24

Table 16.

The dataset arranged in blocks when r = 2.

Block					B	$\frac{B}{k}$	$\frac{B^2}{k}$	B^2
b1	1	8	15
	23.6	22.03	20.98		66.61	22.2	1479	4436.89
b2	1	11	12
	23.74	17.68	27.82		69.24	23.08	1598.1	4794.18
b3	2	10	12
	22.84	16.02	22.29		61.15	20.38	1246.4	3739.32
b4	2	7	15
	23.53	15.25	20.8		59.58	19.86	1183.3	3549.78
b5	3	7	14
	20.79	15.09	16.13		52.01	17.34	901.68	2705.04
b6	3	8	13
	15.87	14.81	16.41		47.09	15.7	739.16	2217.47
b7	4	9	11
	16.36	20.19	16.82		53.37	17.79	949.45	2848.36
b8	4	6	14
	17.52	15.36	17.26		50.14	16.71	838.01	2514.02
b9	5	6	13
	17.68	16.03	17.39		51.1	17.03	870.4	2611.21
b10	5	9	10
	16.71	20.49	16.79		53.99	18	971.64	2914.92
Total					564.3		10,777	32331.2

Table 17.

Block estimation.

T	b1	b2	b3	b4	b5	b6	b7	b8	b9	b10	T	Q	q
1	23.6	23.74									47.34
	22.2	23.08									45.283	2.06	7.66
2			22.84	23.53							46.37
			20.38	19.86							40.243	6.13	2.62
3					20.79	15.87					36.66
					17.34	15.7					33.033	3.63	−4.6
4							16.36	17.52			33.88
							17.79	16.71			34.503	−0.62	−3.1
5									17.68	16.71	34.39
									17.03	18	35.03	−0.64	−2.6
6								15.36	16.03		31.39
								16.71	17.03		33.747	−2.36	−3.9
7				15.25	15.09						30.34
				19.86	17.34						37.197	−6.86	−0.4
8	22.03					14.81					36.84
	22.2					15.7					37.9	−1.06	0.28
9							20.19			20.49	40.68
							17.79			18	35.787	4.89	−1.8
10			16.02							16.79	32.81
			20.38							18	38.38	−5.57	0.76
11		17.68					16.82				34.5
		23.08					17.79				40.87	−6.37	3.25
12		27.82	22.29								50.11
		23.08	20.38								43.463	6.65	5.84
13						16.41			17.39		33.8
						15.7			17.03		32.73	1.07	−4.9
14					16.13			17.26			33.39
					17.34			16.71			34.05	−0.66	−3.6
15	20.98			20.8							41.78
	22.2			19.86							42.063	−0.28	4.44

Table 18.

Treatments estimation.

G	1	2	3		Gi
1	1	6	11
	2.06	−2.36	−6.37		−6.67
2	2	7	12
	6.13	−6.86	6.65		5.92
3	3	8	13
	3.63	−1.06	1.07		3.64
4	4	9	14
	−0.62	4.89	−0.66		3.61
5	5	10	15
	−0.64	−5.57	−0.28		−6.49

Table 19.

Group for treatment adjustment.

T	Qi	Gi	W = $\frac{k}{r\left(k-1\right)}$Qi	Z = $\frac{1}{r\left(k-1\right)}$Gi	T = W-Z	TQi
1	2.06	−6.67	1.545	−1.6675	3.2125	6.61775
2	6.13	5.92	4.5975	1.48	3.1175	19.11028
3	3.63	3.64	2.7225	0.91	1.8125	6.579375
4	−0.62	3.61	−0.465	0.9025	−1.3675	0.84785
5	−0.64	−6.49	−0.48	−1.6225	1.1425	−0.7312
6	−2.36	−6.67	−1.77	−1.6675	−0.1025	0.2419
7	−6.86	5.92	−5.145	1.48	−6.625	45.4475
8	−1.06	3.64	−0.795	0.91	−1.705	1.8073
9	4.89	3.61	3.6675	0.9025	2.765	13.52085
10	−5.57	−6.49	−4.1775	−1.6225	−2.555	14.23135
11	−6.37	−6.67	−4.7775	−1.6675	−3.11	19.8107
12	6.65	5.92	4.9875	1.48	3.5075	23.32488
13	1.07	3.64	0.8025	0.91	−0.1075	−0.11503
14	−0.66	3.61	−0.495	0.9025	−1.3975	0.92235
15	−0.28	−6.49	−0.21	−1.6225	1.4125	−0.3955
Total						151.2204

Table 20.

Treatment adjustment estimates.

(1)	$\frac{G^2}{N}$ = $\frac{564.3x564.3}{30}$	10614.483
(2)	$\sum x^2$	10939.08
(3)	$\frac{\sum B^2}{k}$	10,777

Table 21.

Sum of square table.

Case 1

When r = 2

$$\mathrm{t}=15,\mathrm{k}=3,\mathrm{r}=2,\mathrm{b}=10$$

$$\mathrm{Let}\mathrm{p}=\frac{\mathit{rG}}{N}=\frac{2x564.28}{30}=37.6187$$

Then q₁ = 45.283–37.62 = 7.66

$${\mathrm{Q}}_1=47.34–45.283=2.06$$

$$\mathrm{SSB}=\left(3\right)–\left(1\right)$$

$${\mathrm{SS}}_{\text{Total}}=\left(2\right)–\left(1\right)$$

The model assumption

$${\mathrm{Y}}_{\mathrm{ij}}=\mathrm{\mu }+{\mathrm{\alpha }}_{\mathrm{i}}+{\mathrm{\beta }}_{\mathrm{j}}+{\mathrm{e}}_{\mathrm{ij}}$$

Where α is the treatment, β is the block, e_ij is the error term and μ is the mean.

The Anova Table 22 shows that both the blocks and the treatments adjustment are significant (p-value ˂0.05). Therefore, we can identify the treatments responsible for the significance (Figure 4).

SV	SS	df	MS	F	P-value
blocks	162.517	9	18.0574	9.977	0.00557***
Treatments adj	151.2204	14	10.8015	5.968	0.0186***
Error	10.8596	6	1.8099
Total	324.597	29

Table 22.

ANOVA for the Design when r = 2.

Note: Estimates with “***” means they are significant since the p < 0.05 is considered significant in the study.

Figure 4.

Chart showing the treatment and block means.

The Table 23 shows that treatments 1,2,12 and 15 contributed to the significant of the results (Table 24)

T	Ti	Bi	Mean
1	47.34	45.283	23.67
2	46.37	40.243	23.185
3	36.66	33.033	18.33
4	33.88	34.503	16.94
5	34.39	35.03	17.195
6	31.39	33.747	15.695
7	30.34	37.197	15.17
8	36.84	37.9	18.42
9	40.68	35.787	20.34
10	32.81	38.38	16.405
11	34.5	40.87	17.25
12	50.11	43.463	25.055
13	33.8	32.73	16.9
14	33.39	34.05	16.695
15	41.78	42.063	20.89

Table 23.

Treatment, block and mean.

					Sum
b1	1(23.6)	8(22.03)	15(20.98)		24
b2	1(23.74)	11(17.68)	12(27.82)		24
b3	1(18.39)	10(16.02)	13(16.41)		24
b4	2(22.84)	10(16.79)	12(22.29)		24
b5	2(23.53)	7(15.25)	15(20.8)		24
b6	2(20.95)	8(14.81)	14(16.13)		24
b7	3(20.79)	7(15.09)	14(17.26)		24
b8	3(15.87)	8(14.83	13(17.39)		24
b9	3(13.98)	9(20.19)	12(22.35)		24
b10	4(16.36)	9(20.49)	11(16.82)		24
b11	4(17.52)	6(15.36)	14(21)		24
b12	4(19.07)	5(16.01)	15(14.93)		24
b13	5(17.68)	6(16.03)	13(18.16)		24
b14	5(16.71)	9(21.66)	10(17.8)		24
b15	6(16.26)	7(15.83)	11(16.82)		24

Table 24.

Table showing r = 3.

Case 2

When r = 3, then

$$\mathrm{t}=15,\mathrm{k}=3,\mathrm{r}=3\text{and}\mathrm{b}=15$$

Following the similar calculation, the Anova table was estimate in Table 25

SV	SS	df	MS	F	P-value
Blocks	186.569	14	13.326	2.1896	0.0674
Treatments adj	150.09	14	10.721	1.7616	0.1385
Error	97.373	16	6.086
Total	434.032	44

Table 25.

ANOVA for the design when r = 3.

Table 25 showed that both the blocks and the treatments are not significant since the p-value is greater than 0.05

4. Discussion

This study has introduced Constant Block-Size with Constant Sum-Block Partially Balanced Incomplete Block Designs (CBS-CSB PBIBDs) as a new class of partially balanced incomplete block designs that will be used when there is need to prioritize the treatments combination. This study defined constant block-size; a design is said to have a constant block-size if the number of experimental unit, k, are the same irrespective of the number of blocks, b. Also the study defined constant block-sum; a design is said to be constant block-sum if the number of blocks, b, are the same, $\mathrm{b}=\frac{t}{k}$, and the sum of each corresponding number of experimental units are the same, irrespective of the number of treatments and replicates [24]. It is determined by the sum of all the treatments divided by number of blocks, $\mathrm{CBS}=\frac{\sum t}{b}$. Furthermore, this study gave conditions on constant sum-block; there is constant block size, each corresponding blocks has constant sum, the number of blocks are not the same, $\mathrm{b}=\frac{\mathit{tr}}{k}$, the number of treatments, t, are constant and the number of replicates, r, are at least one, $\mathrm{r}\ge 1$. This study also gave conditions for Constant Block-Size with Constant Sum-Block Partially Balanced Incomplete Block Designs, CBS-CSB PBIBDs; the treatments combinations must be prioritized, that is, there must be number of zero associates, λ₂ = 0, the maximum number of replicates must be necessarily determine from treatment one and the number of treatments and block sizes remain fixed. The study designed t = 21, k = 3; t = 15, k = 3; t = 27, k = 3. And their respective cases were designed and presented with line charts. It was observed when the number of replicate, r, increases the efficiency factors increases. It was also observed that when r = 1, then E = 0.

After the design of the experiments, the study analyzes the experiments using real-life application dataset. The dataset was collected on the impact of Body Mass Index (BMI) on impact oscillometry (IOS) measures on children with sickle cell diseases. [26]. Forty-five (45) observations were extracted which has the following factors; asthma, hydroxyurea, ICS and LABA. Furthermore, each treatment was replicated 3 times alongside with gender. The BMI results were recorded. The dataset was addressed in two cases (2 and 3) based on the priorities. The priority is based on the fixed sum of 24. For the case one, r = 2, the study shows that both the blocks and the treatments adjustment are significant since the p-value is less than 0.05. But the case two, r = 3, the study shows that both the blocks and the treatments are not significant since the p-value is greater than 0.05.

Consequently, the study is not applicable on the even number of treatments, i.e., when t $=$ 2, 4, 6 … Even number of treatments will never give a constant block size [15]. In that case, there is need to research on constant sum-block with even number of treatments.

5. Conclusion

In conclusion, Constant Block-Size with Constant Sum-Block Partially Balanced Incomplete Block Designs (CBS-CSB PBIBDs) has been constructed in this study as a new class of partially balanced incomplete block designs that should be used when it requires prioritizing the treatments combination. The CBS-CSB PBIBDs are designed to provide efficient and precise estimates of treatment effects and they are useful in obtaining maximum information from the available resources, leading to increased experimental efficiency. The CBS-CSB PBIBDs constructed are applicable for testing the impact of various factors on product quality, manufacturing processes, or other parameters of interest. Constant sum-block with even number of treatments will be considered for future research.