Derivation of Two Associate PBIBD from Necessary Properties of BIBD

Troon John Benedict; Onyango Fredrick; Karanjah Anthony

doi:doi:10.11648/j.ajtas.20231204.11

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Derivation of Two Associate PBIBD from Necessary Properties of BIBD

Troon John Benedict, Onyango Fredrick, Karanjah Anthony

Published in American Journal of Theoretical and Applied Statistics (Volume 12, Issue 4)

Received: 23 April 2023 Accepted: 13 May 2023 Published: 6 July 2023

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Abstract

One of the main shortfall of Balanced Incomplete Block Designs is that the design is not available for all parameter sets. Thus, some of the parameter sets that satisfy the necessary conditions for BIBD cannot be constructed as such. This means that the parameter sets for the designs is not possible to be constructed with a single association scheme (λ). Given that the structural difference that exist between BIBD and PBIBD is that the later do not have a single association scheme but rather m association schemes, the study believed that if the BIBD parameter sets could not be constructed as a single associations scheme then maybe if the design is broken down to more than one association scheme in form of a PBIBD then maybe such design might later be constructed using other methods as PBIBDs. The study aimed at determining whether two association scheme PBIBDs could be derived from necessary properties of BIBDs. The main reason for maiking the transformation is that for BIBD some of the designs that satisfy the necessary properties have been determined not to exist meaning that the designs could not be constructed using a single association scheme λ. Therefore, the study felt that if the designs could be broken down into 2 association scheme (λ₁ and λ₂) then such a design might be constructed as a PBIBD. The study related the necessary properties of BIBD and two association scheme PBIBD. Using the properties, the study created eight sets of linear equations and using the Gauss Jordan Elimination method, the study was able to solve for the eight unknown parameters of the PBIBD association scheme. The study was able in the end to convert a BIBD that satisfy necessary properties of BIBD into a two association scheme PBIBD that satisfy all the necessary properties of PBIBD.

Published in	American Journal of Theoretical and Applied Statistics (Volume 12, Issue 4)
DOI	10.11648/j.ajtas.20231204.11
Page(s)	66-71
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), 2023. Published by Science Publishing Group

Keywords

BIBD, PBIBD, Association Scheme, Necessary Properties, Derivation

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]	Arasu, K. T., Bansal, P. and Watson, C. (2013). Partially Balanced Incomplete Block Design with Two Associate Classes. Journal of Statistical Planning and Inference 143; 983–991.
[5]	Bose, R. C. (1939), On the construction of balanced incomplete block designs. Annals of Eugenics, Vol. 9, pp. 353–399.
[6]	Bose, R. C. and Shimamoto, T. (1952). Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Stat. Assoc, Jfl, 151-184.
[7]	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.
[8]	Dey, A. (2010). Incomplete Block Designs. World Scientific Publishing Co. Pte. Ltd. Warren Street. U.S.A.
[9]	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.
[10]	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.
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[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]	Janardan, M. (2018). Construction of balanced incomplete block design: an application of galois field. Open Science Journal of Statistics and Application. 2013; 5 (3): 32-39.
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[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]	Sinha, K. (1987). Generalized Partially Balanced Incomplete Block Designs. Discrete Mathematics 67; 315-318.
[20]	Shih, Y. (1969). Partially Balanced Incomplete Block Designs. All Graduate Plan B and other Reports. 1133. https://digitalcommons.usu.edu/gradreports/1133
[21]	Manohar, N. V. (1959). The non-existence of certain PBIB designs, Ann. Math. Stat. 30, 1051-1062.
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APA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. (2023). Derivation of Two Associate PBIBD from Necessary Properties of BIBD. American Journal of Theoretical and Applied Statistics, 12(4), 66-71. https://doi.org/10.11648/j.ajtas.20231204.11

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ACS Style

Troon John Benedict; Onyango Fredrick; Karanjah Anthony. Derivation of Two Associate PBIBD from Necessary Properties of BIBD. Am. J. Theor. Appl. Stat. 2023, 12(4), 66-71. doi: 10.11648/j.ajtas.20231204.11

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AMA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. Derivation of Two Associate PBIBD from Necessary Properties of BIBD. Am J Theor Appl Stat. 2023;12(4):66-71. doi: 10.11648/j.ajtas.20231204.11

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@article{10.11648/j.ajtas.20231204.11,
  author = {Troon John Benedict and Onyango Fredrick and Karanjah Anthony},
  title = {Derivation of Two Associate PBIBD from Necessary Properties of BIBD},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {4},
  pages = {66-71},
  doi = {10.11648/j.ajtas.20231204.11},
  url = {https://doi.org/10.11648/j.ajtas.20231204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231204.11},
  abstract = {One of the main shortfall of Balanced Incomplete Block Designs is that the design is not available for all parameter sets. Thus, some of the parameter sets that satisfy the necessary conditions for BIBD cannot be constructed as such. This means that the parameter sets for the designs is not possible to be constructed with a single association scheme (λ). Given that the structural difference that exist between BIBD and PBIBD is that the later do not have a single association scheme but rather m association schemes, the study believed that if the BIBD parameter sets could not be constructed as a single associations scheme then maybe if the design is broken down to more than one association scheme in form of a PBIBD then maybe such design might later be constructed using other methods as PBIBDs. The study aimed at determining whether two association scheme PBIBDs could be derived from necessary properties of BIBDs. The main reason for maiking the transformation is that for BIBD some of the designs that satisfy the necessary properties have been determined not to exist meaning that the designs could not be constructed using a single association scheme λ. Therefore, the study felt that if the designs could be broken down into 2 association scheme (λ1 and λ2) then such a design might be constructed as a PBIBD. The study related the necessary properties of BIBD and two association scheme PBIBD. Using the properties, the study created eight sets of linear equations and using the Gauss Jordan Elimination method, the study was able to solve for the eight unknown parameters of the PBIBD association scheme. The study was able in the end to convert a BIBD that satisfy necessary properties of BIBD into a two association scheme PBIBD that satisfy all the necessary properties of PBIBD.},
 year = {2023}
}

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TY  - JOUR
T1  - Derivation of Two Associate PBIBD from Necessary Properties of BIBD
AU  - Troon John Benedict
AU  - Onyango Fredrick
AU  - Karanjah Anthony
Y1  - 2023/07/06
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231204.11
DO  - 10.11648/j.ajtas.20231204.11
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  - 66
EP  - 71
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231204.11
AB  - One of the main shortfall of Balanced Incomplete Block Designs is that the design is not available for all parameter sets. Thus, some of the parameter sets that satisfy the necessary conditions for BIBD cannot be constructed as such. This means that the parameter sets for the designs is not possible to be constructed with a single association scheme (λ). Given that the structural difference that exist between BIBD and PBIBD is that the later do not have a single association scheme but rather m association schemes, the study believed that if the BIBD parameter sets could not be constructed as a single associations scheme then maybe if the design is broken down to more than one association scheme in form of a PBIBD then maybe such design might later be constructed using other methods as PBIBDs. The study aimed at determining whether two association scheme PBIBDs could be derived from necessary properties of BIBDs. The main reason for maiking the transformation is that for BIBD some of the designs that satisfy the necessary properties have been determined not to exist meaning that the designs could not be constructed using a single association scheme λ. Therefore, the study felt that if the designs could be broken down into 2 association scheme (λ1 and λ2) then such a design might be constructed as a PBIBD. The study related the necessary properties of BIBD and two association scheme PBIBD. Using the properties, the study created eight sets of linear equations and using the Gauss Jordan Elimination method, the study was able to solve for the eight unknown parameters of the PBIBD association scheme. The study was able in the end to convert a BIBD that satisfy necessary properties of BIBD into a two association scheme PBIBD that satisfy all the necessary properties of PBIBD.
VL  - 12
IS  - 4
ER  -

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Author Information

Troon John Benedict

Department of Mathematics and Physical Sciences, 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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APA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. (2023). Derivation of Two Associate PBIBD from Necessary Properties of BIBD. American Journal of Theoretical and Applied Statistics, 12(4), 66-71. https://doi.org/10.11648/j.ajtas.20231204.11

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ACS Style

Troon John Benedict; Onyango Fredrick; Karanjah Anthony. Derivation of Two Associate PBIBD from Necessary Properties of BIBD. Am. J. Theor. Appl. Stat. 2023, 12(4), 66-71. doi: 10.11648/j.ajtas.20231204.11

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AMA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. Derivation of Two Associate PBIBD from Necessary Properties of BIBD. Am J Theor Appl Stat. 2023;12(4):66-71. doi: 10.11648/j.ajtas.20231204.11

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@article{10.11648/j.ajtas.20231204.11,
  author = {Troon John Benedict and Onyango Fredrick and Karanjah Anthony},
  title = {Derivation of Two Associate PBIBD from Necessary Properties of BIBD},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {4},
  pages = {66-71},
  doi = {10.11648/j.ajtas.20231204.11},
  url = {https://doi.org/10.11648/j.ajtas.20231204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231204.11},
  abstract = {One of the main shortfall of Balanced Incomplete Block Designs is that the design is not available for all parameter sets. Thus, some of the parameter sets that satisfy the necessary conditions for BIBD cannot be constructed as such. This means that the parameter sets for the designs is not possible to be constructed with a single association scheme (λ). Given that the structural difference that exist between BIBD and PBIBD is that the later do not have a single association scheme but rather m association schemes, the study believed that if the BIBD parameter sets could not be constructed as a single associations scheme then maybe if the design is broken down to more than one association scheme in form of a PBIBD then maybe such design might later be constructed using other methods as PBIBDs. The study aimed at determining whether two association scheme PBIBDs could be derived from necessary properties of BIBDs. The main reason for maiking the transformation is that for BIBD some of the designs that satisfy the necessary properties have been determined not to exist meaning that the designs could not be constructed using a single association scheme λ. Therefore, the study felt that if the designs could be broken down into 2 association scheme (λ1 and λ2) then such a design might be constructed as a PBIBD. The study related the necessary properties of BIBD and two association scheme PBIBD. Using the properties, the study created eight sets of linear equations and using the Gauss Jordan Elimination method, the study was able to solve for the eight unknown parameters of the PBIBD association scheme. The study was able in the end to convert a BIBD that satisfy necessary properties of BIBD into a two association scheme PBIBD that satisfy all the necessary properties of PBIBD.},
 year = {2023}
}

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TY  - JOUR
T1  - Derivation of Two Associate PBIBD from Necessary Properties of BIBD
AU  - Troon John Benedict
AU  - Onyango Fredrick
AU  - Karanjah Anthony
Y1  - 2023/07/06
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231204.11
DO  - 10.11648/j.ajtas.20231204.11
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  - 66
EP  - 71
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231204.11
AB  - One of the main shortfall of Balanced Incomplete Block Designs is that the design is not available for all parameter sets. Thus, some of the parameter sets that satisfy the necessary conditions for BIBD cannot be constructed as such. This means that the parameter sets for the designs is not possible to be constructed with a single association scheme (λ). Given that the structural difference that exist between BIBD and PBIBD is that the later do not have a single association scheme but rather m association schemes, the study believed that if the BIBD parameter sets could not be constructed as a single associations scheme then maybe if the design is broken down to more than one association scheme in form of a PBIBD then maybe such design might later be constructed using other methods as PBIBDs. The study aimed at determining whether two association scheme PBIBDs could be derived from necessary properties of BIBDs. The main reason for maiking the transformation is that for BIBD some of the designs that satisfy the necessary properties have been determined not to exist meaning that the designs could not be constructed using a single association scheme λ. Therefore, the study felt that if the designs could be broken down into 2 association scheme (λ1 and λ2) then such a design might be constructed as a PBIBD. The study related the necessary properties of BIBD and two association scheme PBIBD. Using the properties, the study created eight sets of linear equations and using the Gauss Jordan Elimination method, the study was able to solve for the eight unknown parameters of the PBIBD association scheme. The study was able in the end to convert a BIBD that satisfy necessary properties of BIBD into a two association scheme PBIBD that satisfy all the necessary properties of PBIBD.
VL  - 12
IS  - 4
ER  -

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