Skip to main content
SpringerLink
Log in

Non-orthogonal graeco-latin designs

  • Conference paper
  • First Online:
Combinatorial Mathematics IV

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 560))

Abstract

Statisticians are interested in designs for two non-interacting sets of treatments. These designs present many interesting combinatorial problems. The subject is reviewed from a combinatorial viewpoint, and unsolved problems are indicated. An extensive bibliography is appended.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography 1

  1. K. Afsarinejad and A. Hedayat, Some contributions to the theory of multistage Youden design, Ann. Statistics 3 (1975), 707–711.

    MathSciNet  MATH  Google Scholar 

  2. H.L. Agrawal, A method of construction of three-factor balanced designs, J. Indian Statist. Assoc. 4 (1966), 10–13.

    MathSciNet  Google Scholar 

  3. H.L. Agrawal, Some methods of construction of designs for two-way elimination of heterogeneity. I, J. Amer. Statist. Assoc. 61 (1966), 1153–1171.

    MathSciNet  Google Scholar 

  4. H.L. Agrawal, Some systematic methods of construction of designs for two-way elimination of heterogeneity, Calcutta Statist. Assoc. Bull. 15 (1966), 93–108.

    Article  MathSciNet  MATH  Google Scholar 

  5. H.L. Agrawal and B.S. Boob, A method of construction of 4DIB designs, Calcutta Statist. Assoc. Bull. 23 (1974), 125–128.

    Article  MathSciNet  MATH  Google Scholar 

  6. H.L. Agrawal and R.I. Mishra, Some methods of construction of 4DIB designs, Calcutta Statist. Assoc. Bull. 20 (1971), 89–92.

    Article  MATH  Google Scholar 

  7. D.A. Anderson, Designs with partial factorial balance, Ann. Math. Statist. 43 (1972), 1333–1341.

    Article  MathSciNet  MATH  Google Scholar 

  8. R.C. Bose and J.N. Srivastava, Multidimensional partially balanced designs and their analysis, with applications to partially balanced factorial fractions, Sankhyā A 26 (1964), 145–168.

    MathSciNet  MATH  Google Scholar 

  9. B.D. Causey, Some examples of multi-dimensional incomplete block designs, Ann. Math. Statist. 39 (1968), 1577–1590.

    MathSciNet  MATH  Google Scholar 

  10. G.M. Clarke, A second set of treatments in a Youden square design, Biometrics 19 (1963), 98–104.

    Article  MathSciNet  Google Scholar 

  11. G.M. Clarke, Four-way balanced designs based on Youden squares with 5, 6 or 7 treatments, Biometrics 23 (1967), 803–812.

    Article  MathSciNet  Google Scholar 

  12. G. Dall’Aglio, Blocs incomplets équilibrés orthogonaux, Colloques Internat. C.N.R.S. No. 110 (1963), 195–214.

    Google Scholar 

  13. M.Ja. Èingorin and T.N. Èingorina, Some aspects of the relationship between multiple-coincidence storage devices and the theory of Latin hypersquares. I. (In Russian) Izvestija Vus. Radiofizika 10 (1967), 1569–1575. English translation: Radiophysics and Quantum Electronics 10 (1967), 880–883.

    Google Scholar 

  14. M.Ja. Èingorin and T.N. Èingorina, Some aspects of the relationship between multiple-coincidence storage devices and the theory of Latin hypersquares. II. (In Russian) Izvestija Vus. Radiofizika 10 (1967), 1576–1595. English translation: Radiophysics and Quantum Electronics 10 (1967), 884–893.

    Google Scholar 

  15. M.Ja. Èingorin and T.N. Èingorina, Maximum selection in storage or decoding matrices. (In Russian) Izvestija Vuz. Radiofizika 10 (1967), 1596–1598. English translation: Radiophysics and Quantum Electronics 10 (1967), 894–895.

    Google Scholar 

  16. T.N. Èingorina, On the existence of sets of orthogonal Latin hyper-rectangles. (In Russian) Izvestija Vuz. Radiofizika 12 (1969), 1732–1739. English translation: Radiophysics and Quantum Electronics 12 (1969), 1350–1355.

    MathSciNet  Google Scholar 

  17. T.N. Èingorina and M.Ja. Èingorin, On the existence of complete sets of orthogonal Latin hyper-rectangles. (In Russian) Izvestija Vuz. Radiofizika 12 (1969), 1721–1731. English translation: Radiophysics and Quantum Electronics 12 (1969), 1341–1349.

    MathSciNet  Google Scholar 

  18. T.N. Èingorina and M.Ja. Èingorin, On the existence of sets of stitchings for multidimensional storages and decoders. (In Russian) Izvestija Vuz. Radiofizika 12 (1969), 1740–1745. English translation: Radiophysics and Quantum Electronics 12 (1969), 1356–1359.

    Google Scholar 

  19. W.T. Federer, Construction of classes of experimental designs using transversals in Latin squares and Hedayat’s sum composition method. Statistical Papers in Honor of George W. Snedecor, (ed. by T.A. Bancroft, Ames: Iowa State Univ. Press, 1972, 91–114.

    Google Scholar 

  20. W.T. Federer and D. Raghavarao, On augmented designs, Biometrics 31 (1975), 29–35.

    Article  MathSciNet  MATH  Google Scholar 

  21. D.J. Finney, Some orthogonal properties of the 4×4 and 6×6 Latin squares, Ann. Eugen. 12 (1945), 213–219.

    Article  MathSciNet  MATH  Google Scholar 

  22. D.J. Finney, Orthogonal partitions of the 5×5 Latin squares, Ann. Eugen. 13 (1946), 1–3.

    Article  MathSciNet  MATH  Google Scholar 

  23. D.J. Finney, Orthogonal partitions of the 6×6 Latin squares, Ann. Eugen. 13 (1946), 184–196.

    Article  MathSciNet  MATH  Google Scholar 

  24. G.H. Freeman, Some experimental designs of use in changing from one set of treatments to another, Part I, J.R. Statist. Soc. B 19 (1957), 154–162.

    MATH  Google Scholar 

  25. G.H. Freeman, Some experimental designs for use in changing from one set of treatments to another, Part II: existence of the designs, J.R. Statist. Soc. B 19 (1957), 163–165.

    Google Scholar 

  26. G.H. Freeman, Families of designs for two successive experiments, Ann. Math. Statist. 29 (1958), 1063–1078.

    Article  MathSciNet  MATH  Google Scholar 

  27. G.H. Freeman, The use of the same experimental material for more than one set of treatments, Appl. Statist. 8 (1959), 13–20.

    Article  Google Scholar 

  28. G.H. Freeman, Some further designs of type O:PP, Ann. Math. Statist. 32 (1961), 1186–1190.

    Article  MathSciNet  MATH  Google Scholar 

  29. G.H. Freeman, The addition of further treatments to Latin square designs, Biometrics 20 (1964), 713–729.

    Article  MATH  Google Scholar 

  30. G.H. Freeman, Some non-orthogonal partitions of 4×4, 5×5 and 6×6 Latin squares, Ann. Math. Statist. 37 (1966), 666–681.

    Article  MathSciNet  MATH  Google Scholar 

  31. G.H. Freeman, Experimental designs with many classifications, J.R. Statist. Soc. B 34 (1972), 84–99.

    MathSciNet  MATH  Google Scholar 

  32. G.H. Freeman, Row-and-column designs with two groups of treatments having different replications, J.R. Statist. Soc. B 37 (1975), 114–128.

    MathSciNet  MATH  Google Scholar 

  33. W.B. Hall and E.R. Williams, Cyclic superimposed designs, Biometrika 60 (1973), 47–53.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Hedayat, E.T. Parker and W.T. Federer, The existence and construction of two families of designs for two successive experiments, Biometrika 57 (1970), 351–355.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. Hedayat and D. Raghavarao, 3-way BIB designs, J. Combin. Theory A 18 (1975), 207–209.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Hedayat and E. Seiden, F-square and orthogonal F-squares design; a generalisation of Latin square and orthogonal Latin squares design, Ann. Math. Statist. 41 (1970), 2035–2044.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Hedayat, E. Seiden and W.T. Federer, Some families of designs for multistage experiments: mutually balanced Youden designs when the number of treatments is prime power or twin primes. I, Ann. Math. Statist. 43 (1972), 1517–1527.

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Hedayat and S.S. Shrikhande, Experimental designs and combinatorial systems associated with Latin squares and sets of mutually orthogonal Latin squares, Sankhyā A 33 (1971), 423–432.

    MathSciNet  MATH  Google Scholar 

  39. T.N. Hoblyn, S.C. Pearce and G.H. Freeman, Some considerations in the design of successive experiments in fruit plantations, Biometrics 10 (1954), 503–515.

    Article  Google Scholar 

  40. J. Kiefer, On the nonrandomised optimality and randomised nonoptimality of symmetrical designs, Ann. Math. Statist. 29 (1958), 675–699.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Kiefer, Optimum experimental designs (with discussion), J.R. Statist. Soc. B 21 (1959), 272–319.

    MathSciNet  MATH  Google Scholar 

  42. J. Kiefer, Balanced block designs and generalized Youden designs, I: Construction (patchwork), Ann. Statist. 3 (1975), 109–118.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Lesavre, Peut-on parler de rectangles latins orthogonaux?, Publ. économétr. Fac. Sci. Lyon 1, No. 2 (1968), 65–72.

    MATH  Google Scholar 

  44. S.C. Pearce, The use and classification of non-orthogonal designs, J.R. Statist. Soc. A 126 (1963), 353–377.

    Article  MathSciNet  Google Scholar 

  45. S.C. Pearce and J. Taylor, The changing of treatments in a long-term trial, J. Agric. Sci. 38 (1948), 402–410.

    Article  Google Scholar 

  46. R.F. Potthoff, Three-factor additive designs more general than the Latin square, Technometrics 4 (1962), 187–208.

    Article  MathSciNet  MATH  Google Scholar 

  47. R.F. Potthoff, Four-factor additive designs more general than the Graeco-Latin square, Technometrics 4 (1962), 361–366.

    Article  MathSciNet  MATH  Google Scholar 

  48. R.F. Potthoff, Some illustrations of four-dimensional incomplete block constructions, Calcutta Statist. Assoc. Bull. 12 (1963), 19–30.

    Article  MathSciNet  MATH  Google Scholar 

  49. D.A. Preece, Some row and column designs for two sets of treatments, Biometrics 22 (1966), 1–25.

    Article  MathSciNet  Google Scholar 

  50. D.A. Preece, Some balanced incomplete block designs for two sets of treatments, Biometrika 53 (1966), 497–506.

    Article  MathSciNet  Google Scholar 

  51. D.A. Preece, Balanced 6×6 designs for 9 treatments, Sankhyā B 30 (1968), 443–446.

    Google Scholar 

  52. D.A. Preece, Some new balanced row-and-column designs for two non-interacting sets of treatments, Biometrics 27 (1971), 426–430.

    Article  Google Scholar 

  53. D.A. Preece, Some designs based on 11×5 Youden "squares", Utilitas Math 9 (1976), (to appear).

    Google Scholar 

  54. D.A. Preece and P.J. Cameron, Some new fully-balanced Graeco-Latin Youden "squares", Utilitas Math. 8 (1975), 193–204.

    MathSciNet  MATH  Google Scholar 

  55. D. Raghavarao and G. Nageswararao, A note on a method of construction of designs for two-way elimination of heterogeneity, Communications in Statistics 3 (1974), 197–199.

    Article  MathSciNet  MATH  Google Scholar 

  56. J.N. Srivastava and D.A. Anderson, Some basic properties of multi-dimensional partially balanced designs, Ann. Math. Statist. 41 (1970), 1438–1445.

    Article  MathSciNet  MATH  Google Scholar 

  57. J.N. Srivastava and D.A. Anderson, Factorial association schemes with applications to the construction of multidimensional partially balanced designs, Ann. Math. Statist. 42 (1971), 1167–1181.

    Article  MathSciNet  MATH  Google Scholar 

  58. J.W. Archbold and N.L. Johnson, A construction for Room’s squares and an application in experimental design, Ann. Math. Statist. 29 (1958), 219–225.

    Article  MathSciNet  MATH  Google Scholar 

  59. R.C. Bose and T. Shimamoto, Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Statist. Assoc. 47 (1952), 151–184.

    Article  MathSciNet  MATH  Google Scholar 

  60. J. Dénes and A.D. Keedwell, Latin Squares and Their Applications. (Univ. Press, London, Eng., 1974).

    MATH  Google Scholar 

  61. R.M. Jones, On a property of incomplete blocks, J.R. Statist. Soc. B 21 (1959), 172–179.

    MathSciNet  MATH  Google Scholar 

  62. H.W. Norton, The 7×7 squares, Ann. Eugen. 9 (1939), 269–307.

    Article  MathSciNet  MATH  Google Scholar 

  63. D.A. Preece, Combinatorial analysis and experimental design: a review of "Constructions and Combinatorial Problems in Design of Experiments" by Damaraju Raghavarao, The Statistician 21 (1972), 77–87.

    Article  Google Scholar 

  64. D.A. Preece, S.C. Pearce and J.R. Kerr, Orthogonal designs for three-dimensional experiments, Biometrika 60 (1973), 349–358.

    MathSciNet  MATH  Google Scholar 

  65. W.D. Wallis, Room squares (Part 2 of Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices by W.D. Wallis, Anne P. Street and Jennifer S. Wallis). Lecture Notes in Math., 292. (Springer-Verlag, Berlin, 1972, 29–121.)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Kent at, Canterbury, England

    D. A. Preece

  2. Division of Mathematics and Statistics, C.S.I.R.O., UK

    D. A. Preece

Authors
  1. D. A. Preece
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Louis R. A. Casse Walter D. Wallis

Rights and permissions

Reprints and permissions

Copyright information

© 1976 Springer-Verlag

About this paper

Cite this paper

Preece, D.A. (1976). Non-orthogonal graeco-latin designs. In: Casse, L.R.A., Wallis, W.D. (eds) Combinatorial Mathematics IV. Lecture Notes in Mathematics, vol 560. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097364

Download citation

  • DOI: https://doi.org/10.1007/BFb0097364

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08053-4

  • Online ISBN: 978-3-540-37537-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation