Residual Reduced Balanced Incomplete Block Design

Troon John Benedict; Onyango Fredrick; Karanjah Anthony

doi:doi:10.11648/j.ajtas.20231205.13

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Residual Reduced Balanced Incomplete Block Design

Troon John Benedict, Onyango Fredrick, Karanjah Anthony

Published in American Journal of Theoretical and Applied Statistics (Volume 12, Issue 5)

Received: 11 June 2023 Accepted: 29 June 2023 Published: 14 October 2023

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Abstract

The construction of Balanced Incomplete Block Designs is a combination problem that involves the arrangement of v treatments into b blocks each of size k such that each treatment is replicated exactly r times in the design and a pair of treatments occur together in λ blocks. Researchers have devised a number of methods that can be used in constructing BIBDs, using geometry, difference sets, existing BIBD designs, computers and mathematical algorithms, and Latin squares. However, the existing constructing methods still cannot be used to construct all the BIBDs. This has left the existence of some BIBDs to still be unknown as some of them still cannot be constructed using the existing construction methods. The study aimed to derive a new construction method that uses the un-reduced BIBD to construct a new class of BIBD known as Residual Reduced BIBD. The study used the un-reduced BIBD with parameters (v, k) to construct the new class of BIBD. Consider an un-reduced BIBD with parameters v and k such that k≥3 the Residual Reduced BIBD was derived from the un-reduced design selection of blocks of the un-reduced BIBD that contain a particular treatment i. Then in the selected blocks if treatments i deleted and the rest of the treatments are left, then this forms a BIBD known as Residual Reduced BIBD. Residual Reduced BIBD formed has the parameters v^* = v -1, b^* = ((v - 1)!(v - k))/(k!(v - k)!), k^* = k, r^* = ((v - 2)!(v - k))/((k - 1)!(v - k)!), λ^* = ((v - 3)!(v - k))/((k - 2)!(v - k)!). In conclusion, the study was able to show that a new class of BIBD could be constructed from the un-reduced BIBD. This means that some other BIBDs still can be derived from this universal set using appropriate procedures.

Published in	American Journal of Theoretical and Applied Statistics (Volume 12, Issue 5)
DOI	10.11648/j.ajtas.20231205.13
Page(s)	117-119
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Un-Reduced BIBD, Incomplete Block Design, Treatment, Reduced Residual, Balanced Incomplete Block Design, Construction of BIBD

References

[1]	Akra, U. P., Akpan, S. S., Ugbe, T. A. and Ntekim, O. E. (2021). Finite Euclidean Geometry Approach for Constructing Balanced Incomplete Block Design (BIBD). Asian Journal of Probability and Statistics. 11 (4): 47-59.
[2]	Alabi, M. A. (2018). Construction of balanced incomplete block design of lattice series I and II. International Journal of Innovative Scientific and Engineering Technologies Research. 2018; 6 (4): 10-22.
[3]	Alam, N. M. (2014). On Some Methods of Construction of Block Designs. I. A. S. R. I, Library Avenue, New Delhi-110012.
[4]	Bose, R. C. (1939), On the construction of balanced incomplete block designs. Annals of Eugenics, Vol. 9, pp. 353–399.
[5]	Bose, R. C., Shrikhande, S. S., and Parker, E. T. (1960). Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Canadian Journal of Mathematics, 12, 189-203.
[6]	Bruck, R. H. and Ryser, H. J. (1949), The non-existence of certain finite projective planes. Canadian Journal of Mathematics, Vol. 1, pp. 88-93.
[7]	Fisher, R. A. (1940). An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugenics, 10, 52-75.
[8]	Greig, M., and Rees, D. H. (2003). Existence of balanced incomplete block designs for many sets of treatments. Discrete Mathematics, 261 (1-3), 299-324.
[9]	Goud T. S. and Bhatra, C. N. Ch. (2016). Construction of Balanced Incomplete Block Designs. International Journal of Mathematics and Statistics Invention. 4 (1) 2321-4767.
[10]	Hanani, H. (1961). The existence and construction of balanced incomplete block designs. The Annals of Mathematical Statistics, 32 (2), 361-386.
[11]	Hinkelmann, K. and Kempthorne, O. (2005). Design and Analysis of Experiments. John Wiley and Sons, Inc., Hoboken, New Jersey.
[12]	Hsiao-Lih, J., Tai-Chang, H. and Babul, M. H. (2007). A study of methods for construction of balanced incomplete block design. Journal of Discrete Mathematical Sciences and Cryptography Vol. 10 (2007), No. 2, pp. 227–243.
[13]	Jane di Paola, Jennifer Seberry Wallis and W. D. Wallis, A list of balanced incomplete block designs for r < 30, Proceedings of the Fourth Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, 8, (1973), 249-258.
[14]	Khare, M. and W. T. Federer (1981). A simple construction procedure for resolvable incomplete block designs for any number of treatments. Biom. J., 23, 121-132.
[15]	Mahanta, J. (2018). Construction of balanced incomplete block design: An application of Galois field. Open Science Journal of Statistics and Application.
[16]	Mandal, B. N. (2015). Linear Integer Programming Approach to Construction of Balanced Incomplete Block Designs. Communications in Statistics - Simulation and Computation, 44: 6, 1405-1411, DOI: 10.1080/03610918.2013.821482.
[17]	Montgomery, D. C. (2019). Design and analysis of experiment. John Wiley and Sons, New York.
[18]	Neil, J. S. (2010). Construction of balanced incomplete block design. Journal of Statistics and Probability. 12 (5); 231–343.
[19]	Pachamuthu, A. R. M. (2018). On the construction of balanced incomplete block designs using MOLS of order six - a special case. International Journal of Creative Research Thoughts. 6 (1) 2320-2882.
[20]	Wan, Z. X. (2009). Design theory. World Scientific Publishing Company.
[21]	Yasmin, F., Ahmed, R. and Akhtar, M. (2015). Construction of Balanced Incomplete Block Designs Using Cyclic Shifts. Communications in Statistics—Simulation and Computation 44: 525–532. DOI: 10.1080/03610918.2013.784984.
[22]	Yates, F. (1936). A new method of arranging variety trials involving a large number of varieties. J. Agric. Sci., 26, 424-445.

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APA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. (2023). Residual Reduced Balanced Incomplete Block Design. American Journal of Theoretical and Applied Statistics, 12(5), 117-119. https://doi.org/10.11648/j.ajtas.20231205.13

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ACS Style

Troon John Benedict; Onyango Fredrick; Karanjah Anthony. Residual Reduced Balanced Incomplete Block Design. Am. J. Theor. Appl. Stat. 2023, 12(5), 117-119. doi: 10.11648/j.ajtas.20231205.13

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AMA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. Residual Reduced Balanced Incomplete Block Design. Am J Theor Appl Stat. 2023;12(5):117-119. doi: 10.11648/j.ajtas.20231205.13

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@article{10.11648/j.ajtas.20231205.13,
  author = {Troon John Benedict and Onyango Fredrick and Karanjah Anthony},
  title = {Residual Reduced Balanced Incomplete Block Design},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {5},
  pages = {117-119},
  doi = {10.11648/j.ajtas.20231205.13},
  url = {https://doi.org/10.11648/j.ajtas.20231205.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231205.13},
  abstract = {The construction of Balanced Incomplete Block Designs is a combination problem that involves the arrangement of v treatments into b blocks each of size k such that each treatment is replicated exactly r times in the design and a pair of treatments occur together in λ blocks. Researchers have devised a number of methods that can be used in constructing BIBDs, using geometry, difference sets, existing BIBD designs, computers and mathematical algorithms, and Latin squares. However, the existing constructing methods still cannot be used to construct all the BIBDs. This has left the existence of some BIBDs to still be unknown as some of them still cannot be constructed using the existing construction methods. The study aimed to derive a new construction method that uses the un-reduced BIBD to construct a new class of BIBD known as Residual Reduced BIBD. The study used the un-reduced BIBD with parameters (v, k) to construct the new class of BIBD. Consider an un-reduced BIBD with parameters v and k such that k≥3 the Residual Reduced BIBD was derived from the un-reduced design selection of blocks of the un-reduced BIBD that contain a particular treatment i. Then in the selected blocks if treatments i deleted and the rest of the treatments are left, then this forms a BIBD known as Residual Reduced BIBD. Residual Reduced BIBD formed has the parameters v* = v -1, b* = ((v - 1)!(v - k))/(k!(v - k)!), k* = k, r* = ((v - 2)!(v - k))/((k - 1)!(v - k)!), λ* = ((v - 3)!(v - k))/((k - 2)!(v - k)!). In conclusion, the study was able to show that a new class of BIBD could be constructed from the un-reduced BIBD. This means that some other BIBDs still can be derived from this universal set using appropriate procedures.},
 year = {2023}
}

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TY  - JOUR
T1  - Residual Reduced Balanced Incomplete Block Design
AU  - Troon John Benedict
AU  - Onyango Fredrick
AU  - Karanjah Anthony
Y1  - 2023/10/14
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231205.13
DO  - 10.11648/j.ajtas.20231205.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 117
EP  - 119
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231205.13
AB  - The construction of Balanced Incomplete Block Designs is a combination problem that involves the arrangement of v treatments into b blocks each of size k such that each treatment is replicated exactly r times in the design and a pair of treatments occur together in λ blocks. Researchers have devised a number of methods that can be used in constructing BIBDs, using geometry, difference sets, existing BIBD designs, computers and mathematical algorithms, and Latin squares. However, the existing constructing methods still cannot be used to construct all the BIBDs. This has left the existence of some BIBDs to still be unknown as some of them still cannot be constructed using the existing construction methods. The study aimed to derive a new construction method that uses the un-reduced BIBD to construct a new class of BIBD known as Residual Reduced BIBD. The study used the un-reduced BIBD with parameters (v, k) to construct the new class of BIBD. Consider an un-reduced BIBD with parameters v and k such that k≥3 the Residual Reduced BIBD was derived from the un-reduced design selection of blocks of the un-reduced BIBD that contain a particular treatment i. Then in the selected blocks if treatments i deleted and the rest of the treatments are left, then this forms a BIBD known as Residual Reduced BIBD. Residual Reduced BIBD formed has the parameters v* = v -1, b* = ((v - 1)!(v - k))/(k!(v - k)!), k* = k, r* = ((v - 2)!(v - k))/((k - 1)!(v - k)!), λ* = ((v - 3)!(v - k))/((k - 2)!(v - k)!). In conclusion, the study was able to show that a new class of BIBD could be constructed from the un-reduced BIBD. This means that some other BIBDs still can be derived from this universal set using appropriate procedures.
VL  - 12
IS  - 5
ER  -

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Author Information

Troon John Benedict

Department of Mathematics and Physical Science, Maasai Mara University, Narok, Kenya
Onyango Fredrick

Department of Mathematics and Actuarial Science, Maseno University, Luanda, Kenya
Karanjah Anthony

Department of Mathematics, Multimedia University, Nairobi, Kenya

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Troon John Benedict, Onyango Fredrick, Karanjah Anthony. (2023). Residual Reduced Balanced Incomplete Block Design. American Journal of Theoretical and Applied Statistics, 12(5), 117-119. https://doi.org/10.11648/j.ajtas.20231205.13

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ACS Style

Troon John Benedict; Onyango Fredrick; Karanjah Anthony. Residual Reduced Balanced Incomplete Block Design. Am. J. Theor. Appl. Stat. 2023, 12(5), 117-119. doi: 10.11648/j.ajtas.20231205.13

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AMA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. Residual Reduced Balanced Incomplete Block Design. Am J Theor Appl Stat. 2023;12(5):117-119. doi: 10.11648/j.ajtas.20231205.13

Copy | Download

@article{10.11648/j.ajtas.20231205.13,
  author = {Troon John Benedict and Onyango Fredrick and Karanjah Anthony},
  title = {Residual Reduced Balanced Incomplete Block Design},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {5},
  pages = {117-119},
  doi = {10.11648/j.ajtas.20231205.13},
  url = {https://doi.org/10.11648/j.ajtas.20231205.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231205.13},
  abstract = {The construction of Balanced Incomplete Block Designs is a combination problem that involves the arrangement of v treatments into b blocks each of size k such that each treatment is replicated exactly r times in the design and a pair of treatments occur together in λ blocks. Researchers have devised a number of methods that can be used in constructing BIBDs, using geometry, difference sets, existing BIBD designs, computers and mathematical algorithms, and Latin squares. However, the existing constructing methods still cannot be used to construct all the BIBDs. This has left the existence of some BIBDs to still be unknown as some of them still cannot be constructed using the existing construction methods. The study aimed to derive a new construction method that uses the un-reduced BIBD to construct a new class of BIBD known as Residual Reduced BIBD. The study used the un-reduced BIBD with parameters (v, k) to construct the new class of BIBD. Consider an un-reduced BIBD with parameters v and k such that k≥3 the Residual Reduced BIBD was derived from the un-reduced design selection of blocks of the un-reduced BIBD that contain a particular treatment i. Then in the selected blocks if treatments i deleted and the rest of the treatments are left, then this forms a BIBD known as Residual Reduced BIBD. Residual Reduced BIBD formed has the parameters v* = v -1, b* = ((v - 1)!(v - k))/(k!(v - k)!), k* = k, r* = ((v - 2)!(v - k))/((k - 1)!(v - k)!), λ* = ((v - 3)!(v - k))/((k - 2)!(v - k)!). In conclusion, the study was able to show that a new class of BIBD could be constructed from the un-reduced BIBD. This means that some other BIBDs still can be derived from this universal set using appropriate procedures.},
 year = {2023}
}

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TY  - JOUR
T1  - Residual Reduced Balanced Incomplete Block Design
AU  - Troon John Benedict
AU  - Onyango Fredrick
AU  - Karanjah Anthony
Y1  - 2023/10/14
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231205.13
DO  - 10.11648/j.ajtas.20231205.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 117
EP  - 119
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231205.13
AB  - The construction of Balanced Incomplete Block Designs is a combination problem that involves the arrangement of v treatments into b blocks each of size k such that each treatment is replicated exactly r times in the design and a pair of treatments occur together in λ blocks. Researchers have devised a number of methods that can be used in constructing BIBDs, using geometry, difference sets, existing BIBD designs, computers and mathematical algorithms, and Latin squares. However, the existing constructing methods still cannot be used to construct all the BIBDs. This has left the existence of some BIBDs to still be unknown as some of them still cannot be constructed using the existing construction methods. The study aimed to derive a new construction method that uses the un-reduced BIBD to construct a new class of BIBD known as Residual Reduced BIBD. The study used the un-reduced BIBD with parameters (v, k) to construct the new class of BIBD. Consider an un-reduced BIBD with parameters v and k such that k≥3 the Residual Reduced BIBD was derived from the un-reduced design selection of blocks of the un-reduced BIBD that contain a particular treatment i. Then in the selected blocks if treatments i deleted and the rest of the treatments are left, then this forms a BIBD known as Residual Reduced BIBD. Residual Reduced BIBD formed has the parameters v* = v -1, b* = ((v - 1)!(v - k))/(k!(v - k)!), k* = k, r* = ((v - 2)!(v - k))/((k - 1)!(v - k)!), λ* = ((v - 3)!(v - k))/((k - 2)!(v - k)!). In conclusion, the study was able to show that a new class of BIBD could be constructed from the un-reduced BIBD. This means that some other BIBDs still can be derived from this universal set using appropriate procedures.
VL  - 12
IS  - 5
ER  -

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