Abstract
We discuss issues pertaining to existence or nonexistence of triangular designs with constant block-sum property as previously introduced by R. Khattree. The problem has two aspects: the quantitative treatments must be equally spaced or treatments do not have to be equally spaced, and the levels of these treatments are to be determined by the design itself. The incidence matrix and the eigen properties of the corresponding concurrence matrix play a critical role in this investigation. In the process of developing results, we also observe connections between constant block-sum property of the partially balanced incomplete block designs with some associated balanced incomplete block designs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bansal N, Garg DK (2020) Construction and existence of constant block sum pbib designs. Commun Stat Theory Methods 51(7):2231–2241. https://doi.org/10.1080/03610926.2020.1772308
Bose RC, Shimamoto T (1952) Classification and analysis of partially balanced incomplete block designs with two associate classes. J Am Stat Assoc 47(258):151–184
Chai FS, Das A, Midha CK (2013) Construction of magic rectangles of odd order. Australas J Comb 55:131–144
Chai FS, Singh R, Stufken J (2019) Nearly magic rectangles. J Comb Des 27(9):562–567
Chang L-C, Liu CW, Liu WR (1965) Incomplete block designs with triangular parameters for which \(k\)\(\le \) 10 and \(r\)\(\le \) 10. Sci Sinica 14(3):329–338
Clatworthy WH (1955) Partially balanced incomplete block designs with two associate classes and two treatments per block. J Res Natl Bur Stand 54(2):177–190
Clatworthy WH (1956) Contributions on partially balanced incomplete block designs with two associate classes, volume 47. US Government Printing Office
Clatworthy WH (1973) Tables of two-associate-class partially balanced designs, volume 63. US National Bureau of Standards
De Los Reyes JP, Das A (2007) A matrix approach to construct magic rectangles of even order. Technical Report Number isid/ms/2007/07. Indian Statistical Institute, New Delhi
Dey A (1986) Theory of block designs. Wiley, New Yok
Gupta S (2021) Constant block-sum group divisible designs. Stat Appl 19(1):141–148
Hall M, Connor WS (1954) An embedding theorem for balanced incomplete block designs. Can J Math 6:35–41
Huilgol MI, Asok S (2022) \(PBIB\)-designs with constant block sums arising from different graph parameters
John PWM (1998) Statistical design and analysis of experiments. SIAM
Khattree R (2015) Mixture experiments in the interior: Yantram designs. J Stat Theory Pract 9(4):797–822. https://doi.org/10.1080/15598608.2015.1029148
Khattree R (2016) A class of designs for mixture experiments implied by the numerical configurations in Hindu/Jain yantrams. J Indian Stat Assoc 54(1–2):127–155
Khattree R (2019) A note on the nonexistence of the constant block-sum balanced incomplete block designs. Commun Stat Theory Methods 48(20):5165–5168. https://doi.org/10.1080/03610926.2018.1508715
Khattree R (2019) The Parshvanath yantram yields a constant-sum partially balanced incomplete block design. Commun Stat Theory Methods 48(4):841–849. https://doi.org/10.1080/03610926.2017.1417439
Khattree R (2020) On construction of constant block-sum partially balanced incomplete block designs. Commun Stat Theory Methods 49(11):2585–2606. https://doi.org/10.1080/03610926.2019.1576895
Khattree R (2022) Constant and nearly constant block-sum partially balanced incomplete block designs and magic rectangles. Revision submitted in Journal of Statistical Theory and Practice
Khattree R (2022) On construction of equireplicated constant block-sum designs. Commun Stat Theory Methods 51(13):4434–4450. https://doi.org/10.1080/03610926.2020.1814816
Mandal BN (2021) On incomplete block designs with constant block sums [paper presentation]. In: Annual conference on visionary innovations in statistical theory and applications–2021 (VISTA 2021), India, 2021. [During Feb 24-28, 2021]
Raghavarao D (1971) Constructions and combinatorial problems in design of experiments. Wiley
Shrikhande SS (1960) Relations between certain incomplete block designs, contributions to probability and statistics. pp 388–395
Shrikhande SS (1965) On a class of partially balanced incomplete block designs. Ann Math Stat 36:1807–1814
Funding
This work has not been externally funded by any organization other than authors’ respective employers. Part of the results in this paper are included in the PhD dissertation of the first author.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Ethical Statements
The material in this article is the authors’ own original work, which has not been previously published elsewhere, and all authors have consented to its publication. There are no conflicting or competing financial or nonfinancial interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A.1: Constant Block-Sum Version of Srikhande’s Design for \(m =9\)
Appendix A.2: Fourteen Orthogonal Vectors Needed for Ragavarao’s Design for \(m =7\)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bansal, N., Garg, D.K. & Khattree, R. Some Results on the Existence or Nonexistence of Constant Block-sum Triangular and Other Related Designs. J Stat Theory Pract 17, 1 (2023). https://doi.org/10.1007/s42519-022-00301-8
Accepted:
Published:
DOI: https://doi.org/10.1007/s42519-022-00301-8