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Incomplete Block Designs for Comparing Test Treatments with Multiple Controls

Vinayaka1,2, B.N. Mandal3, Rajender Parsad4, Vinaykumar L.N.1,*, P. Murali2, Amaresh2, Shweta Kumari2
1The Graduate School, ICAR-Indian Agricultural Research Institute, Pusa, New Delhi- 110 012, India.
2ICAR-Sugarcane Breeding Institute, Coimbatore-641 007, Tamil Nadu, India.
3ICAR-Indian Agricultural Research Institute, Hazaribagh-825 405, Jharkhand, India.
4ICAR-Indian Agricultural Statistics Research Institute, Pusa, New Delhi-110 012, India.

  • Submitted07-03-2024|

  • Accepted06-05-2024|

  • First Online 18-06-2024|

  • doi 10.18805/BKAP723

ABSTRACT

Background: This article provides some new construction methods of partially balanced bipartite block (PBBB) designs for comparing test treatments with more than one control.

Methods: Partially balanced incomplete block (PBIB) designs based on some association schemes such as group divisible association, and cyclic association are used for developing these methods of construction. 

Result: A catalogue of efficient PBBB designs is included for parameter values n1 (number of test treatments) ≤10, n2 (number of control treatments) ≤10 and r2 (replications of test treatments) and (replications of control treatments) ≤15 along with computed variances using SAS code.

REFERENCES


  1. Bechhofer, R.E. and Tamhane, A.C. (1981). Incomplete block designs for comparing treatments with a control: General theory. Technometrics. 23(1): 45-57.

  2. Cheng, C.S., Majumdar, D., Stufken, J. and Türe, T.E. (1988). Optimal step-type designs for comparing test treatments with a control. Journal of the American Statistical Association. 83(402): 477-482.

  3. Clatworthy, W.H. (1973). Tables of two-associate-class partially balanced designs. National Bureau of Standards, Applied Mathematics, Series No. 63, Washington D.C.

  4. Corsten, L.C.A. (1962). Balanced block designs with to different numbers of replicates. Biometrics. 18(4): 499-519.

  5. Das, A., Dey, A., Kageyama, S. and Sinha, K. (2005). A-efficient balanced treatment incomplete block designs. Australasian Journal of Combinatorics. 32: 243-252.

  6. Gupta, V.K. and Parsad, R. (2001). Block designs for comparing test treatments with control treatments-an overview. Statistics and Applications. 3: 133-146.

  7. Harun, M., Varghese, C., Jaggi, S., Varghese, E., Bhowmik, A., Datta, A. and Kumar, N. (2016b). Designs for breeding trials involving triallel crosses. Bhartiya Krishi Anushandhan Patrika. 31(2): 158-160.

  8. Hedayat, A.S. and Majumdar, D. (1984). A-optimal incomplete block designs for test treatment - control comparisons. Technometrics. 26(4): 363-370.

  9. Hedayat, A.S., Jacroux, M. and Majumdar, D. (1988). Optimal designs for comparing test treatments with controls. Statistical Science. 3(4): 462-491.

  10. Jacroux, M. (1986). On the determination and construction of MV- Optimal block designs for comparing block designs with a standard treatment. Journal of Statistical Planning and Inference. 15: 205-225.

  11. Jacroux, M. (1987). Some MV-Optimal block designs for comparing test treatments with a standard treatment. Sankhya. B49: 239-261.

  12. Jacroux, M. (1988). Some further results on the MV-Optimality of block designs for comparing test treatments to a standard treatment. Journal of Statistical Planning and Inference. 20: 201-214.

  13. Jacroux, M. and Majumdar, D. (1989). Optimal block designs for comparing test treatments with a control when k > v Journal of Statistical Planning and Inference. 23(3): 381-396.

  14. Jaggi, S., Gupta, V.K. and Parsad, R. (1996). A-efficient block designs for comparing two disjoint sets of treatments. Communications in Statistics: Theory and Methods. 25(5): 967-983.

  15. Kageyama, S. and Sinha, K. (1988).  Some constructions of balanced bipartite block designs. Utilitas Mathematica. 33: 137-162.

  16. Kageyama, S. and Sinha, K. (1991). Constructions of partially balanced bipartite block designs. Discrete Mathematics. 92(1-3): 137-144.

  17. Karmakar, S., Varghese, C., Jaggi, S., Harun, M., Kumar, D. (2022). Partially Balanced 3-Designs using Mutually Orthogonal Latin Squares. Bhartiya Krishi Anusandhan Patrika. 37(1): 8-12. DOI: 10.18805/BKAP351.

  18. Majumdar, D. (1996). Optimal and efficient treatment-control designs. Handbook of Statistics. 13: 1007-1053.

  19. Majumdar, D. and Notz, W.I. (1983). Optimal incomplete block designs for comparing treatments with a control. The Annals of Statistics. 258-266.

  20. Mandal, B.N., Gupta, V.K. and Parsad, R. (2017). Balanced treatment incomplete block designs through integer programming. Communications in Statistics-Theory and Methods. 46(8): 3728-3737.

  21. Mandal, B.N., Parsad, R. and Dash, S. (2018). A-optimal block designs for comparing test treatments with control treatment (s)-an algorithmic approach. Project report, IASRI publication, New Delhi.

  22. Mandal, B.N., Parsad, R. and Dash, S. (2020). Construction of A- optimal balanced treatment incomplete block designs: An algorithmic approach. Communications in Statistics- Simulation and Computation. 49(6): 1653-1664.

  23. Notz, W.I. and Tamhane, A. (1983). Balanced treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k=3, p=3 (1) 10. Communications in Statistics-Theory and Methods. 12: 1391-1412.

  24. Parsad, R., Gupta, V.K. and Prasad, N.S.G. (1995). On construction of A-efficient balanced test treatment incomplete block- designs. Utilitas Mathematica. 47: 185-190.

  25. Parsad, R., Gupta, V.K. and Srivastava, R. (2007a). Designs for cropping systems research. Journal of Statistical Planning and Inference. 137: 1687-1703.

  26. Parsad, R., Kageyama, S. and Gupta, V.K. (2007b). Use of complementary property of block designs in PBIB designs. Ars Combinatoria. 85: 173-182.

  27. Puri, P.D. and Kageyama, S. (1985). Constructions of partially efficiency -balanced designs and their analysis. Communications in Statistics-Theory and Methods. 14(6): 1315-1342.

  28. Puri, P.D., Mehta, B.D. and Kageyama, S. (1986). Patterned constructions of partially efficiency-balanced designs. Journal of Statistical Planning and Inference. 15: 365-378.

  29. Rao, M.B. (1966). Partially balanced block designs with two different  number[s] of replications. Journal of Indian Statistical Association. 4: 1-9.

  30. Sinha, K. and Kageyama, S. (1990).  Further constructions of balanced bipartite block designs.  Utilitas Mathematica. 38: 155-160.

  31. Stufken, J. (1987). A-optimal block designs for comparing test treatments with a control. Annals of Statistics. 15: 1629-1638.

  32. Stufken, J. (1988). On bounds for the efficiency of block designs for comparing test treatments with a control. Journal of Statistical Planning and Inference. 19(3): 361-372.

  33. Stufken, J. (1991). On group divisible treatment designs for comparing test treatments with a standard treatment in blocks of size 3. Journal of Statistical Planning and Inference. 28(2): 205-221.

  34. Varghese, C., Jaggi S., Harun, M., and Kumar, D. (2020). Three- associate class partially balanced incomplete block designs through kronecker product. Bhartiya Krishi Anusandhan Patrika. 35(1): 102-105. DOI: 10.18805/ BKAP211.

  35. Vinayaka and Parsad, R. (2023). Resolvable and 2-replicate PBIB designs based on higher association schemes using Polyhedra. Statistics and Applications. 21(2):  141-154.

  36. Vinayaka, Parsad, R., Mandal, B.N., Dash, S., Vinaykumar, L.N., Kumar, M. and Singh, D.R. (2023). Partially balanced bipartite block designs. Communication in Statistics- Theory and Methods. https://doi.org/10.1080/03610926.2023.2251623.

  37. Vinayaka, R. Parsad, and B. N. Mandal (2024). Partially balanced nested block designs based on 2-associate-class association schemes for test-control comparisons. Communications in Statistics-Simulation and Computation. DOI: 10.1080/ 03610918.2024.2317872.

  38. Vinaykumar L.N., Varghese, C., Jaggi, S., Varghese, E., Harun, M., Karmakar, S., Kumar, D. (2023). A Method of Constructing p-rep Designs. Bhartiya Krishi Anusandhan Patrika. 38(3): 223-226. DOI: 10.18805/BKAP561.
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